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Applications of Differentiation

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Applications of Differentiation

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Applications of Differentiation

Tracking the Sign of the Derivative​

If ff is a function, then the sign of its derivative, f′f', indicates whether ff is increasing (f′>0f'>0), decreasing (f′<0f'<0), or zero. +If the derivative takes the value 0 at a certain point x0x_0 then the function has maximum, minimum, or a saddle point at x0x_0.

Details​

If ff is a function, then the sign of its derivative, f′f', indicates whether ff is increasing (f′>0f'>0), decreasing (f′<0f'<0), or zero. +f′f' can be zero at points where ff has a maximum, minimum, or a saddle point.

If f′(x)>0f'(x)>0 for x<x0x< x_0, f′(x0)=0f'(x_0)=0 and f′(x)<0f'(x)<0 for x>x0x>x_0 then ff has a maximum at x0x_0.

If f′(x)<0f'(x)<0 for x<x0x< x_0, f′(x0)=0f'(x_0)=0 and f′(x)>0f'(x)>0 for x>x0x>x_0 then ff has a minimum at x0x_0.

If f′(x)>0f'(x)>0 for x<x0x< x_0, f′(x0)=0f'(x_0)=0 and f′(x)>0f'(x)>0 for x<x0x< x_0 then ff has a saddle point at x0x_0.

If f′(x)<0f'(x)<0 for x<x0x< x_0, f′(x0)=0f'(x_0)=0 and f′(x)<0f'(x)<0 for x<x0x< x_0 then ff has a saddle point at x0x_0.

Examples​

Example

If ff is a function such that its derivative is given by

f′(x)=(x−1)(x−2)(x−3)(x−4),f'(x) = (x-1)(x-2)(x-3)(x-4),

then applying the above criteria for maxima and minima, we see that ff has maxima at 11 and 33 and ff has minima at 22 and 44.

Describing Extrema Using f′′f''​

x0x_0 with f′(x0)=0f'(x_0)=0 corresponds to a maximum if f′′(x0)<0f''(x_0)<0 - x0x_0

x0x_0 with f′(x0)=0f'(x_0)=0 corresponds to a minimum if f′′(x0)>0f''(x_0)>0

Details​

If f′(x0)=0f'(x_0)=0 corresponds to a maximum, then the derivative is decreasing and the second derivative cannot be positive, (i.e. f′′(x0)≤0f''(x_0)\leq 0). +In particular, if the second derivative is strictly negative, ( f′′(x0)<0f''(x_0) <0 ), then we are assured that the point is indeed a maximum, and not a saddle point.

If f′(x0)=0f'(x_0)=0 corresponds to a minimum, then the derivative is increasing and the second derivative cannot be negative, (i.e. f′′(x0)≥0f''(x_0) \geq 0).

If the second derivative is zero, then the point may be a saddle point, as happens with f(x)=x3f(x)=x^3 at x=0x=0.

The Likelihood Function​

If pp is the probability mass function (p.m.f.):

p(x)=P[X=x]p(x) = P [X = x]

then the joint probability of obtaining a sequence of outcomes from independent sampling is:

p(x1)⋅p(x2)⋅p(x3)…p(xn)p(x_1) \cdot p(x_2) \cdot p(x_3) \ldots p(x_n)

Suppose each probability includes some parameter θ\theta, this is written:

pθ(x1),…pθ(xn){p_{\theta}}(x_1), \ldots {p_{\theta}}(x_n)

If the experiment gives x1,x2…,xnx_1, x_2 \ldots, x_n we can write the probability as a function of the parameters:

Lx(θ)=pθ(x1),…,pθ(xn)L_{\mathbf{x}}(\theta) = p_{\theta}(x_1), \ldots, p_{\theta}(x_n)

This is the likelihood function.

Details​

Definition

Recall that the probability mass function (p.m.f) is a function giving the probability of outcomes of an experiment.

We typically denote the p.m.f. by pp so p(x)p(x) gives the probability of a given outcome, xx, of an experiment. +The p.m.f. commonly depends on some parameter. +We often write

p(x)=P[X=x]p(x) = P [X = x]

If we take a sample of independent measurements, from pp, then the joint probability of a given set of numbers is:

p(x1)⋅p(x2)⋅p(x3)⋅…⋅p(xn)p(x_1) \cdot p(x_2) \cdot p(x_3) \cdot \ldots \cdot p(x_n)

Suppose each probability includes the same parameter θ\theta, then this is typically written:

pθ(x1),…,pθ(xn){p_{\theta}}(x_1), \ldots, {p_{\theta}}(x_n)

Now consider the set of outcomes x1,x2…,xnx_1, x_2 \ldots, x_n from the experiment. +We can now take the probability of this outcome as a function of the parameters.

Definition

Lx(θ)=pθ(x1),…,pθ(xn)L_{\mathbf{x}}(\theta) = p_{\theta}(x_1), \ldots, p_{\theta}(x_n)

This is the likelihood function and we often seek to maximize it to estimate the unknown parameters.

Examples​

Example

Suppose we toss a biased coin nn independent times and obtain xx heads, we know the probability of obtaining xx heads is:

(nx)px(1−p)n−x\displaystyle\binom{n}{x}p^x (1-p)^{n-x}

The parameter of interest is pp and the likelihood function is:

L(p)=(nx)px(1−p)n−xL(p) = \displaystyle\binom{n}{x}p^x (1-p)^{n-x}

If pp is unknown we sometimes wish to maximize this function with respect to pp in order to estimate the true probability pp.

Plotting the Likelihood​

missing slide -- want to give a numeric example and plot LL

Examples​

missing example -- want to give a numeric example and plot LL

Maximum Likelihood Estimation​

If LL is a likelihood function for a p.m.f. pθp_{\theta}, then the value θ^\hat{\theta} which gives the maximum of LL:

L(θ^)=max⁡θ(Lθ)L(\hat{\theta}) = \max_\theta ({L}_\theta)

is the maximum likelihood estimator (MLE) of θ\theta.

Details​

Definition

If LL is a likelihood function for a p.m.f. pθp_{\theta}, then the value θ^\hat{\theta} which gives the maximum of LL :

L(θ^)=max⁡θ(Lθ)L (\hat{\theta}) = \max_\theta ({L}_\theta)

is the maximum likelihood estimator of θ\theta.

Examples​

Example

If xx is the number of heads from nn independent tosses of a coin, the likelihood function is:

Lx(p)=(nx)px(1−p)n−xL_x(p) = \displaystyle{n \choose x} p^x (1-p)^{n-x}

Maximizing this is equivalent to maximizing the logarithm of the likelihood, since logarithmic functions are increasing. +The log-likelihood can be written as:

ℓ(p)=ln⁡(L(p))=ln⁡(nx)+xln⁡(p)+(n−x)ln⁡(1−p)\ell(p) = \ln (L(p))= \ln \displaystyle\binom{n}{x} + x \ln (p) + (n-x) \ln (1-p)

To find possible maxima, we need to differentiate this formula and set the derivative to zero:

0=dℓdp=0+xp+n−x1−p(−1)0 = \displaystyle\frac{d \ell }{dp} = 0 + \displaystyle\frac{x}{p}+\displaystyle\frac{n-x}{1-p}(-1)

0=p(1−p)(x)p−p(1−p)n−x1−p0 = p(1-p) \displaystyle\frac{(x)}{p} - p(1-p) \displaystyle\frac{n-x}{1-p}

0=(1−p)x−p(n−x)0 = (1-p)x - p(n-x)

0=x−px−pn+px=x−pn0 = x - px -pn + px = x-pn

So:

0=x−pn0 = x-pn

p=xnp = \displaystyle\frac{x}{n}

is the extreme and so we can write:

p^=xn\hat{p} = \displaystyle\frac{x}{n}

for the MLE.

Least Squares Estimation​

Least squares: Estimate the parameters θ\theta by minimizing:

∑i=1n(yi−gi(θ))2\displaystyle\sum_{i=1}^{n}{(y_i - g_i (\theta))^2}

Details​

Suppose we have a model linking data to parameters. +In general we are predicting yiy_i as gig_i (θ\theta).

In this case it makes sense to estimate parameters θ\theta by minimizing:

∑i=1n(yi−gi(θ))2\displaystyle\sum_{i=1}^{n}{(y_i - g_i (\theta))^2}

Examples​

Example

One may predict numbers, xix_i, as a mean, μ\mu, plus error. +Consider the simple model xi=μ+ϵix_i = \mu + \epsilon_i, where μ\mu is an unknown parameter (constant) and ϵi\epsilon_i is the error in measurement when obtaining the ithi^{th} observations, xix_i, i=1,…,ni=1,\ldots, n.

A natural method to estimate the parameter is to minimize the squared deviations:

min⁡μ∑i=1n(x−μ)2\min_{\mu} \displaystyle\sum_{i=1}^n \left (x - \mu \right )^2

It is not hard to see that the μ^\hat{\mu} that minimizes this is the mean:

μ^=xˉ\hat{ \mu} = \bar{x}

Example

One also commonly predicts data y1,⋯ ,yny_1, \cdots,y_n with values on a straight line, i.e. with α+βxi\alpha + \beta x_i, where x1,…,xnx_1, \ldots, x_n are fixed numbers. +This leads to the regression problem of finding parameter values for α^\hat{\alpha} and β^\hat{\beta} which gives the best fitting straight line in relation to least squares:

min⁡α,β∑(yi−(α+βxi))2\min_{\alpha,\beta} \displaystyle\sum \left ( y_i - ( \alpha + \beta x_i) \right ) ^2

Example

As a general exercise in finding the extreme of a function, let's look at the function f(θ)=∑i=1N(xiθ−3)2f(\theta)=\displaystyle\sum_{i=1}^N(x_i\theta -3)^2 where xix_i are some constants. +We wish to find the θ\theta that minimizes this sum. +We simply differentiate θ\theta to obtain:

f′(θ)=∑i=1n2(xiθ−3)x1=2∑i=1nxi2θ−2∑i=1n3xif'(\theta) = \displaystyle\sum_{i=1}^n 2(x_i\theta -3)x_1 = 2\displaystyle\sum_{i=1}^n x^2_i\theta - 2\displaystyle\sum_{i=1}^n 3x_i

Thus:

f′(θ)=2θ∑i=1nxi2−2∑i=1n3xi=0⇔θ=∑i=1n3xi∑i=1nxi2\begin{aligned} f'(\theta) &= 2\theta \displaystyle\sum_{i=1}^n x^2_i-2\displaystyle\sum_{i=1}^n 3x_i=0 \\ & \Leftrightarrow \theta = \displaystyle\frac{\displaystyle\sum_{i=1}^n 3x_i}{\displaystyle\sum_{i=1}^n x^2_i} \end{aligned}
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Continuity and Limits

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Continuity and Limits

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Continuity and Limits

The Concept of Continuity​

A function is continuous if it has no jumps. +Thus, small changes in each x0x_0, the input, correspond to small changes in the output, f(x0)f(x_0).

Fig. 15

Figure: The above figure is an example of linear growth.

Thomas Robert Malthus (1766-1834) warned about the dangers of uninhibited population growth.

Details​

A function is said to be discontinuous if it has jumps. +The function is continuous if it has no jumps. +Thus, for a continuous function, small changes in each x0x_0, the input, correspond to small changes in the output, f(x0)f(x_0).

Note

Note that polynomials are continuous as are logarithms (for positive numbers).

Discrete Probabilities and Cumulative Distribution Functions​

The cumulative distribution function for a discrete random variable is discontinuous.

Fig. 16

Details​

Definition

If XX is a random variable with a discrete probability distribution and the probability mass function of

p(x)=P[X=x]p(x)=P[X=x]

then the cumulative distribution function, defined by

F(X)=P[X≤x]F(X)=P[X\leq x]

is discontinuous, i.e. it jumps at points in which a positive probability occurs.

Note

When drawing discontinuous functions, it is common practice to use a filled circle at (x,f(x))(x,f(x)) to clarify what the function value is at a point xx of discontinuity.

Examples​

Example

If a coin is tossed three independent times and XX denotes the number of heads, then XX can only take on the values 00, 11, 22 and 33. +The probability of landing exactly xx heads, P(X=x)P(X=x), is p(x)=(nx)pn(1−p)n−xp(x) = \displaystyle\binom{n}{x} p^n (1-p)^{n-x}. +The probabilities are:

xp(x)F(x)0181813848238783181\begin{array}{c c c} \hline \\ x & p(x) & F(x) \\ \hline \\ 0 & \displaystyle\frac{1}{8} & \displaystyle\frac{1}{8} \\ 1 & \displaystyle\frac{3}{8} & \displaystyle\frac{4}{8} \\ 2 & \displaystyle\frac{3}{8} & \displaystyle\frac{7}{8} \\ 3 & \displaystyle\frac{1}{8} & 1 \\ \hline \end{array}

The cumulative distribution function, F(x)=P[X≤x]=∑t≤xp(t)F(x)=P[X \leq x] = \displaystyle\sum_{t\leq x} p(t) has jumps and is therefore discontinuous.

Note

Notice on the above figure how the circles are filled in, the solid circles indicate where the function value is.

Notes on Discontinuous Function​

A function is discontinuous for values or ranges of the variable that do not vary continuously as the variable increases. +In other words, breaks or jumps.

Fig. 17

Figure: f(x)=1xf(x) = \displaystyle\frac{1}{x}, where x≠0x\neq 0

Details​

A function can be discontinuous in a number of different ways. +Most commonly, it may jump at certain points or increase without bound in certain places.

Consider the function ff, defined by f(x)=1/xf(x)= 1/x when x≠0x\neq 0. +Naturally, 1/x1/x is not defined for x=0x=0. +This function increases towards +∞+\infty as xx goes to zero from the right but decreases to −∞-\infty as xx goes to zero from the left. +Since the function does not have the same limit from the right and the left, it cannot be made continuous at x=0x=0 even if one tries to define f(0)f(0) as some number.

Continuity of Polynomials​

All polynomials, p(x)=a0+a1x+a2x2+…+anxnp(x)=a_0+a_1x+a_2x^2+\ldots +a_n x^n are continuous.

Fig. 18

Details​

It is easy to show that simple polynomials such as p(x)=xp(x)=x, p(x)=a+bxp(x)=a+bx, p(x)=ax2+bx+cp(x)=ax^2+bx+c are continuous functions.

It is generally true that a polynomial of the form

p(x)=a0+a1x+a2x2+…+anxnp(x)=a_0+a_1x+a_2x^2+\ldots +a_n x^n

is a continuous function.

Simple Limits​

A limit is used to describe the value that a function or sequence approaches as the input or index approaches some value. +Limits are used to define continuity, derivatives and integrals.

Fig. 19

Figure: f(x)=xxf(x) = x^x, for x>0x>0

Details​

Definition

A limit describes the value that a function or sequence approaches as the input or index approaches some value.

Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives and integrals.

Consider a function f(x)f(x) and a point x0{x}_0. +If f(x)f(x) gets steadily closer to some number cc as xx gets closer to a number x0x_0, then cc is called the limit of f(x)f(x) as xx goes to x0x_0 and is written as:

c=lim⁡x→x0f(x)c = \lim_{x\to x_0}f(x)

If c=f(x0)c = f(x_0) then ff is continuous at x0x_0.

Examples​

Example

Evaluate the limit of f(x)=x2−16x−4f(x) = \displaystyle\frac{x^{2}-16}{x-4} when x→4x\rightarrow 4, or

lim⁡x→4x2−16x−4\lim_{x\rightarrow 4} \displaystyle\frac{x^{2}-16}{x-4}

Notice that in principle we cannot simply stick in the value x=4x=4 since we would then get 0/00/0 which is not defined. +However we can look at the numerator and try to factor it. +This gives us:

x2−16x−4=(x−4)(x+4)x−4=x+4\displaystyle\frac{x^{2}-16}{x-4} = \displaystyle\frac{(x-4)(x+4)}{x-4} = x+4

and the result has the obvious limit of 4+4=84+4=8 as x→4x\to 4.

Example

Consider the function:

g(x)=1xg(x) = \displaystyle\frac{1}{x}

where xx is a positive real number. +As xx increases, g(x)g(x) decreases, approaching 00 but never getting there since 1x=0\displaystyle\frac{1}{x}=0 has no solution. +One can therefore say: "The limit of g(x)g(x), as xx approaches infinity, is 00," and write:

lim⁡x→∞g(x)=0\lim_{x\to\infty} g(x)=0

More on Limits​

Limits impose a certain range of values that may be applied to the function.

Fig. 20

Figure: The function f(x)=11+e−xf(x)= \displaystyle\frac{1}{1+e^{-x}}

Examples​

Example

The Beverton-Holt stock recruitment curve is given by:

R=αS1+SKR=\displaystyle\frac{\alpha S}{1+\displaystyle\frac{S}{K}}

where α,K>0\alpha, K >0 are constants and SS is biomass and RR is recruitment

The behavior of this curve as SS increases S→∞S\rightarrow\infty is

lim⁡S→∞αS1+SK=αK\lim_{S\to\infty}\displaystyle\frac{\alpha S}{1+\displaystyle\frac{S}{K}} =\alpha K

This is seen by rewriting the formula as follows:

lim⁡S→∞αS1+SK=lim⁡S→∞α1S+1K=αK\lim_{S\to\infty}\displaystyle\frac{\alpha S}{1+\displaystyle\frac{S}{K}} = \lim_{S\to\infty}\displaystyle\frac{\alpha }{\displaystyle\frac{1}{S}+\displaystyle\frac{1}{K}} =\alpha K

Example

A popular model for proportions is:

f(x)=11+e−xf(x) = \displaystyle\frac{1}{1+e^{-x}}

As x increases, e−xe^{-x} decreases which implies that the term 1+e−x1+e^{-x} decreases and hence 11+e−x\displaystyle\frac{1}{1+e^{-x}} increases, from which it follows that ff is an increasing function.

Notice that f(0)=12f(0)=\displaystyle\frac{1}{2} and further,

lim⁡x→∞f(x)=1\lim_{x\to\infty} f(x) = 1

This is seen from considering the components. +Since e−x=1exe^{-x} = \displaystyle\frac{1}{e^{x}} and the exponential function goes to infinity as x→∞x\to\infty, e−xe^{-x} goes to 00 and hence f(x)f(x) goes to 1.

Through a similar analysis one finds that:

lim⁡x→−∞f(x)=0\lim_{x\to-\infty} f(x)=0

since, as x→∞x\rightarrow \infty, first −x→∞-x\rightarrow \infty and second e−x→∞e^{-x} \rightarrow \infty.

Example

Evaluate the limit of f(x)=x+4−2xf(x) = \displaystyle\frac{\sqrt{x + 4} - 2}{x} as x→0x \to 0, i.e. solve:

lim⁡x→0x+4−2x\lim_{x \to 0} \displaystyle\frac{\sqrt{x + 4} - 2}{x}

Since there is an xx in the denominator we cannot just direct substitute the 0 as xx sunce that would give us 00\displaystyle\frac{0}{0}, which is an indeterminate form. +We must do some algebra first. +The square root makes this a little bit tricky. +The way to get rid of the radical is to multiply the numerator by the conjugate.

x+4−2x⋅x+4+2x+4+2\displaystyle\frac{\sqrt{x + 4} - 2}{x} \cdot \displaystyle\frac{\sqrt{x + 4} + 2}{\sqrt{x + 4} + 2}

This gives us:

(x+4)2+2(x+4)−2(x+4)−4x(x+4+2)\displaystyle\frac{(\sqrt{x + 4})^2 + 2(\sqrt{x+4}) - 2(\sqrt{x+4}) -4}{x(\sqrt{x + 4} + 2)}

The numerator reduces to xx, and the xx's will cancel out, leaving us with:

1x+4+2\displaystyle\frac{1}{\sqrt{x + 4} + 2}

At this point we can direct substitute 0 for xx, which will give us:

10+4+2\displaystyle\frac{1}{\sqrt{0 + 4} + 2}

Therefore:

lim⁡x→0x+4−2x=14\lim_{x \to 0} \displaystyle\frac{\sqrt{x + 4} - 2}{x} = \displaystyle\frac{1}{4}

One-sided Limits​

f(x)f(x) may tend towards different numbers depending on whether xx approaches x0x_0 from left or right, usually written:

x→x0+x \rightarrow x_{0+} (from the right)

x→x0−x \rightarrow x_{0-} (from the left).

Fig. 21

Details​

Sometimes a function is such that f(x)f(x) tends to different numbers depending on whether x→x0x \rightarrow x_0 from the right ( x→x0+x \rightarrow x_{0+} ) or from the left (x→x0−x \rightarrow x_{0-}).

If

lim⁡x→x0+f(x)=f(x0)\lim_{x \to x_{0+}} f(x)=f(x_0)

then we say that ff is continuous from the right at x0x_0. +Same thing goes for the limit from the left. +In order for the limit to exist at the point (that is the overall limit, regardless og direction) then it must hold true that

lim⁡x→x0−=lim⁡xtox0+\lim_{x \to x_{0-}} = \lim_{x to x_{0+}}

i.e., the limit is the same from both directions.

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Derivatives

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Derivatives

The Derivative As a Limit​

The derivative of the function ff at the point xx is defined as:

lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \displaystyle\frac{f(x+h) - f(x)}{h}

if this limit exists.

Details​

Definition

The derivative of the function ff at the point xx is defined as

lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \displaystyle\frac{f(x+h) -f(x)}{h}

if this limit exists.

When we write y=f(x)y = f(x), we commonly use the notation dydx\displaystyle\frac{dy}{dx} or f′(x)f'(x) to denote the derivative.

The Derivative of f(x)=a+bxf(x)=a+bx​

If

$f(x)=a+bxf(x) = a + bx

then

f(x+h)=a+b(x+h)=a+bx+bhf(x + h) = a + b(x + h) = a + bx + bh

and thus

lim⁡h→0f(x+h)−f(x)h=lim⁡h→0bhh=b\lim_{h \to 0} \displaystyle\frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \displaystyle\frac{bh}{h}=b

Fig. 27

Details​

If

f(x)=a+bxf(x) = a + bx

then

f(x+h)=a+b(x+h)=a+bx+bhf(x + h) = a + b(x + h) = a + bx + bh

and thus

lim⁡h→0f(x+h)−f(x)h=lim⁡h→0bhh=b\lim_{h \to 0} \displaystyle\frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \displaystyle\frac{bh}{h}=b

Thus

f′(x)=bf'(x)=b

The Derivative of f(x)=xnf(x)=x^n​

If

f(x)=xnf(x)=x^n

then

f′(x)=nxn−1f'(x)=nx^{n-1}

Details​

Let f(x)=xnf(x)=x^n, where nn is a positive integer. +To calculate f′f' we use the binomial theorem in the third step:

f(x+h)−f(x)h=(x+h)n−xnh=∑q=0n−1(nq)xqhn−qh=∑q=0n−1(nq)xqhn−q−1=(nn−1)xn−1=nxn−1\begin{aligned} \displaystyle\frac{f(x+h)-f(x)}{h} &= \displaystyle\frac{(x+h)^n-x^n}{h} \\ & = \displaystyle\frac{\displaystyle\sum_{q=0}^{n-1}\displaystyle\binom{n}{q}x^qh^{n-q}}{h} \\ & = \displaystyle\sum_{q=0}^{n-1}\displaystyle\binom{n}{q}x^qh^{n-q-1} \\ & = \displaystyle\binom{n}{n-1}x^{n-1} \\ & = nx^{n-1} \end{aligned}

Thus, we obtain f′(x)=nxn−1f'(x)=nx^{n-1}.

The Derivative of Ln and Exp​

If

f(x)=exf(x) = e^x

then

f′(x)=exf'(x) = e^x

If

g(x)=ln⁡(x)g(x) = \ln(x)

then

g′(x)=1xg'(x) = \displaystyle\frac{1}{x}

Details​

The derivatives of the exponential function is the exponential function itself. +That is, if

f(x)=exf(x) = e^x

then

f′(x)=exf'(x) = e^x

The derivatives of the natural logarithm, ln⁡(x)\ln(x), is 1x\displaystyle\frac{1}{x}. +That is, if

g(x)=ln⁡(x)g(x) = \ln(x)

then

g′(x)=1xg'(x) = \displaystyle\frac{1}{x}

The Derivative of a Sum and Linear Combination​

If ff and gg are functions then the derivative of f+gf+g is given by f′+g′f' + g'.

Details​

Similarly, the derivative of a linear combination is the linear combination of the derivatives. +If ff and gg are functions and k(x)=af(x)+bg(x)k(x)=af(x) + bg(x) then k′(x)=af′(x)+bg′(x)k'(x)=af'(x)+ bg'(x).

Examples​

Example

If

f(x)=2+3x,g(x)+x3f(x) = 2+3x, g(x)+x^3

then we know that

f′(x)=3,g(x)=3x2f'(x)=3, g(x)=3x^2

and if we write

h(x)=f(x)+g(x)=2+3x+x3h(x)=f(x)+g(x)=2+3x+x^3

then

h′(x)=3+3x2h'(x)=3+3x^2

The Derivative of a Polynomial​

The derivative of a polynomial is the sum of the derivatives of the terms of the polynomial.

Details​

If

p(x)=a0+a1x+⋯+anxnp(x) = a_0+a_1x+\dots +a_n x^n

then

p′(x)=a1+2a2x+3a3x2+4a4x3+⋯+nanx(n−1)p'(x) = a_1+2a_2x+3a_3x^2+4a_4x^3+\dots +na_n x^{(n-1)}

Examples​

Example

If

p(x)=2x4+x3p(x)=2x^4+x^3

then

p′(x)=2dx4dx+dx3dx=2⋅4x3+3x2=8x3+3x2p'(x)=2\displaystyle\frac{dx^4}{dx}+\displaystyle\frac{dx^3}{dx}=2 \cdot 4x^3 +3x^2 = 8x^3 +3x^2

The Derivative of a Product​

If

h(x)=f(x)⋅g(x)h(x)=f(x)\cdot g(x)

then

h′(x)=f′(x)⋅g(x)+f(x)⋅g′(x)h'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x)

Details​

Consider two functions, ff and gg and their product, hh :

h(x)=f(x)⋅g(x)h(x)=f(x)\cdot g(x)

The derivative of the product is given by

h′(x)=f′(x)⋅g(x)+f(x)⋅g′(x)h'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x)

Examples​

Example

Suppose the function ff is given by

f(x)=xex+x2ln⁡xf(x)=xe^x+x^2\ln x

Then the derivative can be computed step by step as

f(x)=dxdxex+xdexdx+dx2dxln⁡x+x2dln⁡xdx=1⋅ex+x⋅ex+2x⋅ln⁡x+x2⋅1x=ex(1+x)+2xln⁡x+x\begin{aligned} f(x) &= \displaystyle\frac{dx}{dx}e^x+x\displaystyle\frac{de^x}{dx}+\displaystyle\frac{dx^2}{dx}\ln x +x^2\displaystyle\frac{d \ln x}{dx} \\ & = 1\cdot e^x + x \cdot e^x + 2x \cdot \ln x + x^2 \cdot \displaystyle\frac{1}{x} \\ & = e^x \left ( 1+x \right ) + 2x \ln x + x \end{aligned}

Derivatives of Composite Functions​

If ff and gg are functions and h=f∘gh=f \circ g so that\ h(x)=f(g(x))h(x) = f(g(x)) then h′(x)=dh(x)dx=f′(g(x))g′(x)h'(x) = \displaystyle\frac{dh(x)}{dx} = f'(g(x)) g'(x)

Examples​

Example

For fixed xx consider:

f(p)=ln⁡(px(1−p)n−x)=ln⁡px+ln⁡(1−p)n−x=xln⁡p+(n−x)ln⁡(1−p)\begin{aligned} f(p) &= \ln(p^{x} (1-p)^{n-x}) \\ &= \ln p^{x} + \ln(1-p)^{n-x} \\ &= x \ln p + (n-x) \ln (1-p) \end{aligned}

Then the derivative is computed as follows:

f′(p)=x1p+n−x1−p(−1)=xp−n−x1−p\begin{aligned} f'(p) &= x \displaystyle\frac{1}{p} + \displaystyle\frac{n-x}{1-p}(-1) \\ &= \displaystyle\frac{x}{p} - \displaystyle\frac{n-x}{1-p} \end{aligned}
Example

For fixed xx and yy consider

f(b)=(y−bx)2f(b) = (y-bx)^2

Then the derivative is computed as follows:

f′(b)=2(y−bx)(−x)=−2x(y−bx)=(−2xy)+(2x2)b\begin{aligned} f'(b) &= 2 (y-bx) (-x) \\ &= -2x (y-bx) \\ &= (-2xy) + (2x^2)b \end{aligned}
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Functions of Functions and the Exponential Function

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Functions of Functions and the Exponential Function

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Functions of Functions and the Exponential Function

Exponential Growth and Decline​

Exponential growth is typically expressed as

y(t)=Aekty(t)=Ae^{kt}

Fig. 12

Figure: Exponential growth curve

Details​

Definition

Exponential growth is the rate of population increase across time when a population is devoid of limiting factors (i.e. competition, resources, etc.) and experiences a constant growth rate.

Exponential growth is typically expressed as:

y(t)=Aekty(t)=Ae^{kt}

where

  • A (sometimes denoted P) = initial population size
  • k = growth rate
  • t =number of time intervals
Note

Note that exponential growth occurs when k>0k>0 and exponential decline occurs when k<0k<0.

The Exponential Function​

An exponential function is a function with the form: f(x)=bxf(x)=b^x

Details​

For the exponential function f(x)=bxf(x)=b^x, xx is a positive integer and bb is a fixed positive real number. +The equation can be rewritten as:

f(x)=bx=b⋅b⋅b⋅⋯⋅bf(x)=b^x=b\cdot b \cdot b \cdot \dots \cdot b

When the exponential function is written as f(x)=exf(x)=e^x then, it has a growth rate at time xx equivalent to the value of exe^x for the function at xx.

Properties of the Exponential Function​

Recall that the methods of the basic arithmetic implies that:

ea+b=eaebe^{a+b} = e^a e^b

for any real numbers aa and bb.

Functions of Functions​

Details​

Consider two functions, ff and gg, each defined for some set of real numbers. +Where xx can be solved in function ff using Y=f(x)Y = f(x) when g(Y)g(Y) exists for all such resulting yy. +If y=f(x)y = f(x) and g(y)g(y) exist then we can compute g(f(x))g(f(x)) for any xx.

If

f(x)=x2 and g(y)=eyf(x) = {x}^2 \text{ and } g(y)= {e}^y

then

g(f(x))=ef(x)=ex2g(f(x))= {e}^{f(x)} = {e}^{x^2}

If we call the resulting function hh, then h(x)=g(f(x))h(x) = g(f(x)). +Another common notation for this is

h=g∘fh = g\circ f

Examples​

Example

If g(x)=3+2xg(x)= {3}+ {2}x and f(x)=5x2f(x) = {5}{x}^2, then

g(f(x))=3+2f(x)=3+10x2,g(f(x)) = {3} +{2} f(x) = {3} +{10x}^2,

and

f(g(x))=5(g(x))2=5(3+2x)2=45+60x+20x2f(g(x)) = {5}{(g(x))}^2 = {5}{({3}+{2x})}^2 = {45}+{60x}+{20x}^2

Storing and Using R Code​

As R code gets more complex (more lines) it is usually stored in files. +Functions are typically stored in separate files.

Examples​

Example

Save the following file (test.r):

x=4
y=8
cat("x+y is", x+y, "\n")

To read the file use:

source("test.r")

and the outcome of the equation is displayed in R.

Storing and Calling Functions In R​

To save a function in a separate file use a command of the form function.r.

Examples​

Example
f<-function(x) { return (exp(sum(x))) }

can be stored in a file function.r and subsequently read using the source command.

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Integrals and Probability Density Functions

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Integrals and Probability Density Functions

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Integrals and Probability Density Functions

Area Under a Curve​

The area under a curve between x=ax=a and x=bx=b (for a positive function) is called the integral of the function.

Fig. 28

Figure: Example 1, 2 and 3

Details​

Definition

The area under a curve between x=ax=a and x=bx=b (for a positive function) is called the integral of the function and is denoted: ∫abf(x)dx\int_{a}^{b} f(x)dx when it exists.

The Antiderivative​

Given a function ff, if there is another function FF such that F′=fF'=f, we say that FF is the antiderivative of ff. +For a function ff the antiderivative is denoted by ∫fdx\int f dx.

Note that if FF is one antiderivative of ff and CC is a constant, then G=F+CG=F+C is also an antiderivative. +It is therefore customary to always include the constant, e.g. ∫xdx=12x2+C\int x dx=\displaystyle\frac{1}{2}x^2+C.

Examples​

Example

The antiderivative of xx to a power raises the power by one and divides it by the new power:

∫xndx=1n+1xn+1+C\int x^n dx=\displaystyle\frac{1}{n+1}x^{n+1} +C

Example

∫exdx=ex+C\int e^x dx=e^x+C

Example

∫1xdx=ln⁡(x)+C\int \displaystyle\frac{1}{x} dx=\ln(x)+C

Example

∫2xex2dx=ex2+C\int 2xe^{x^2} dx=e^{x^2}+C

The Fundamental Theorem of Calculus​

If ff is a continuous function, and F′(x)=f(x)F'(x)=f(x) for x∈[a,b]x\in[a,b], then ∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx=F(b)-F(a)

Detail​

It is not too hard to see that the area under the graph of a positive function ff on the interval [a,b][a,b] must be equal to the difference of the values of its antiderivative at aa and bb. +This also holds for functions which take on negative values and is formally stated below.

Definition

Fundamental theorem of calculus: If FF is the antiderivative of the continuous function ff, i.e. F′=fF'=f for x∈[a,b]x\in[a,b], then ∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx=F(b)-F(a).

This difference is often written as ∫abfdx\int_a^b f dx or [F(x)]ab[F(x)]_a ^b.

Examples​

Example

The area under the graph of xnx^n between 00 and 33 is ∫03xndx=[1n+1xn+1]03=1n+13n+1−1n+10n+1=3n+1n+1\int_0^3 x^n dx = [\displaystyle\frac{1}{n+1}x^{n+1}]_0 ^3=\displaystyle\frac{1}{n+1}3^{n+1}-\displaystyle\frac{1}{n+1}0^{n+1}=\displaystyle\frac{3^{n+1}}{n+1}

Example

The area under the graph of exe^x between 33 and 44 is ∫34exdx=[ex]34=e4−e3\int_3^4 e^x dx =[e^x]_3 ^4= e^4-e^3

Example

The area under the graph of 1x\displaystyle\frac{1}{x} between 11 and aa is ∫1a1xdx=[ln⁡(x)]1a=ln⁡(a)−ln⁡(1)=ln⁡(a)\int_1^a \displaystyle\frac{1}{x} dx =[\ln(x)]_1 ^a= \ln(a)-\ln(1)=\ln(a)

Density Functions​

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.).

Fig. 29

Details​

Definition

If XX is a random variable such that

P(a≤X≤b)=∫abf(x)dxP(a\leq X\leq b)=\int\limits^{b}_{a}f(x)dx

for some function ff which satisfies f(x)≥0f(x)\geq0 for all xx and

∫−∞∞f(x)dx=1\int\limits^\infty_{-\infty} f(x)dx = 1

then ff is said to be a probability density function (p.d.f.) for XX.

Definition

The function

F(x)=∫−∞xf(t)dtF(x)= \int\limits^{x}_{-\infty} f(t)dt

is the cumulative distribution function (c.d.f.).

Examples​

Example

Consider a random variable XX from the uniform distribution, denoted by X∼Unf(0,1)X\sim Unf(0,1). +This distribution has density:

f(x)={1if 0≤x≤10e.w.f(x) = \begin{cases} 1 &\text{if } 0 \leq x \leq 1 \\ 0 &\text{e.w.} \end{cases}

The cumulative distribution function is given by:

P[X≤x]=∫−∞xf(t)dt={0if x<0xif 0≤x≤11elseP[X\leq x] = \int\limits^{x}_{-\infty} f(t)dt = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x \leq 1 \\ 1 & \text{else} \end{cases}
Example

Suppose X∼P(λ)X \sim P(\lambda), where XX may denote the number of events per unit time. +The p.m.f. of XX is described by

p(x)=P[X=x]=e−λλxx! for x=0,1,2,…p(x)=P[X=x]=e^{-\lambda}\displaystyle\frac{\lambda^x}{x!} \text{ for } x = 0,1,2,\dots

Let TT denote the waiting time between events, or simply until the first event. +Consider the event T>tT>t for some number t>0t>0. +If X∼p(λ)X\sim p(\lambda) denotes the number of events per unit time, then let XtX_t be the number of events during the time period for 00 through tt. +Then it is natural to assume Xt∼P(λt)X_t \sim P(\lambda t) and it follows that T>tT>t if and only if Xt=0X_t=0 and we obtain P[T>t]=P[Xt=0]=e−λtP[T>t]=P[X_t=0]=e^{-\lambda t}. +It follows that the c.d.f. of TT is

FT(t)=P[T≤t]=1−P[T>t]=1−e−λt for t>0F_T(t)=P[T\leq t]=1-P[T>t]=1-e^{-\lambda t} \text{ for } t > 0

The p.d.f. of TT is therefore

fT(t)=FT′(t)=ddtFT(t)=ddt(1−e−λt)=0−e−λt⋅(−λ)=λe−λtf_T(t)=F_T'(t)=\displaystyle\frac{d}{dt}F_T(t)=\displaystyle\frac{d}{dt}(1-e^{-\lambda t})=0-e^{- \lambda t} \cdot (-\lambda)=\lambda e^{-\lambda t}

for t≥0t \geq 0 and fT(t)=0 for t<0f_T(t)=0 \text{ for } t < 0

The resulting density

f(t)={λe−λtfor t≥00for t<0f(t) = \begin{cases} \lambda e^{-\lambda t} & \text{for } t \geq0 \\ 0 & \text{for } t<0 \end{cases}

describes the exponential distribution.

This distribution has the expected value

E[T]=∫−∞∞tf(t)dt=λ∫0∞te−λtdtE[T]=\int_{-\infty}^{\infty} tf(t)dt=\lambda \int_{0}^{\infty} t e^{-\lambda t}dt

Let's use integration by parts (see below), i.e.: ∫fg′=fg−∫f′g\int fg' = fg - \int f'g to solve that integral. +Let f=tf=t and g′=e−λtg'=e^{-\lambda t}. +Then f′=1f' = 1 and g=−e−λtλg=-\displaystyle\frac{e^{- \lambda t}}{\lambda}. +We obtain:

=λ([−te−λtλ]0∞−∫0∞−−e−λtλdt)=λ((0−0)−[e−λtλ2]0∞)=−λ(0−1λ2)=1λ\begin{aligned} &= \lambda \left( \left[ \displaystyle\frac{-te^{-\lambda t}}{\lambda}\right]_{0}^{\infty} - \int_0^\infty - \displaystyle\frac{-e^{-\lambda t}}{\lambda} dt \right) \\ &= \lambda \left( (0 - 0) - \left[ \displaystyle\frac{e^{-\lambda t}}{\lambda^2} \right]_0^\infty \right) \\ &= -\lambda \left(0 - \displaystyle\frac{1}{\lambda^2}\right) \\ &= \displaystyle\frac{1}{\lambda} \end{aligned}

Probabilities In R: The Normal Distribution​

R has functions to compute values of probability density functions (p.d.f.) and cumulative distribution functions (c.m.d.) for most common distributions.

Details​

The p.d.f. for the normal distribution is

p(t)=12πe−t22p(t)=\displaystyle\frac{1}{\sqrt{2\pi}}e^{-\displaystyle\frac{t^2}{2}}

The c.d.f. for the normal distribution is

Φ(x)=∫−∞x12πe−t22dt\Phi(x)=\int_{-\infty}^x\displaystyle\frac{1}{\sqrt{2\pi}}e^{-\displaystyle\frac{t^2}{2}}dt

Examples​

Example

dnorm() gives the value of the normal p.d.f.

Example

pnorm() gives the value of the normal c.d.f.

Some Rules of Integration​

Examples​

Example

Using integration by parts we obtain:

∫ln⁡(x)xdx=12x2ln⁡(x)−∫12x2⋅1xdx=12x2ln⁡(x)−∫12xdx=12x2ln⁡(x)−14x2\int \ln(x)x dx= \displaystyle\frac{1}{2}x^2\ln(x)-\int \displaystyle\frac{1}{2}x^2\cdot \displaystyle\frac{1}{x} dx = \displaystyle\frac{1}{2}x^2\ln(x)-\int \displaystyle\frac{1}{2}x dx=\displaystyle\frac{1}{2}x^2\ln(x)-\displaystyle\frac{1}{4}x^2

Example

Consider ∫122xex2dx\int_1^2 2xe^{x^2} dx. +By setting x=g(t)=tx=g(t)=\sqrt{t} we obtain

∫122xex2dx=∫142tet12tdt=∫14etdt=e4−e\int_1^2 2xe^{x^2} dx = \int_1^4 2\sqrt{t}e^{t}\displaystyle\frac{1}{2\sqrt{t}}dt=\int_1^4 e^t dt=e^4-e

Handout​

The two most common "tricks" applied in integration are a) integration by parts and b) integration by substitution

a) Integration by parts

(fg)′=f′g+fg′(fg)'=f'g+fg'

by integrating both sides of the equation we obtain:

fg=∫f′gdx+∫fg′dx↔∫fg′dx=fg−∫f′gdxfg=\int f'g dx + \int fg' dx \leftrightarrow \int fg' dx=fg-\int f'g dx

b) Integration by substitution

Consider the definite integral ∫abf(x)dx\int_a^b f(x) dx and let gg be a one-to-one differential function for the interval (c,d)(c,d) to (a,b)(a,b). +Then

∫abf(x)dx=∫cdf(g(y))g′(y)dy\int_a^b f(x) dx=\int_c^d f(g(y))g'(y) dy

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Inverse Functions and the Logarithm

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Inverse Functions and the Logarithm

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Inverse Functions and the Logarithm

Inverse Function​

If ff is a function, then the function gg is the inverse function of ff if

g(f(x))=xg(f(x))=x

for all xx in which f(x)f(x) can be calculated.

Details​

The inverse of a function ff is denoted by f−1f^{-1}, i.e.

f−1(f(x))=xf^{-1}(f(x))=x

Examples​

Example

If f(x)=x2f(x) = x^2 for x<0x<0 then the function gg, defined as g(y)=yg(y)=\sqrt{y} for y>0y>0, is not the inverse of ff since g(f(x))=x2=∣x∣=−xg(f(x))=\sqrt{x^2} = |x| = -x for x<0x<0.

When the Inverse Exists: The Domain Question​

Inverses do not always exist. +For an inverse of ff to exist, ff must be one-to-one, i.e. for each xx, f(x)f(x) must be unique.

Fig. 13

Figure: The function f(x)=x2f(x)=x^2 does not have an inverse since f(x)=1f(x)=1

has two possible solutions −1-1 and 11.

Examples​

Example

f(x)=x2f(x)=x^2 does not have an inverse since f(x)=1f(x)=1 has two possible solutions -1 and 1.

Note

Note that iff ff is a function, then the function gg is the inverse function of ff, if g(f(x))=xg(f(x)) = x for all calculated values of xx in f(x)f(x).

The inverse function of ff is denoted by f−1f^{-1}, i.e. f−1(f(x))=xf^{-1}(f(x)) = x.

Example

What is the inverse function, f−1f^{-1}, of ff if f(x)=5+4xf(x) = 5 + 4x. +The simplest approach is to write y=f(x)y=f(x) and solve for xx.

With

f(x)=5+4xf(x) = 5 + 4x

we write

y=5+4xy = 5 + 4x

which we can now rewrite as

y−5=4xy - 5 = 4x

and this implies

y−54=x\displaystyle\frac{y-5}{4} = x

And there we have it, very simple:

f−1(f(x))=y−54f^{-1}(f(x)) = \displaystyle\frac{y - 5}{4}

The Base 10 Logarithm​

When xx is a positive real number in x=10yx=10^y, yy is referred to as the base 10 logarithm of x and is written as:

y=log⁡10(x)y=\log_{10}(x)

or

y=log⁡(x)y=\log(x)

Details​

If log⁡(x)=a\log (x) = a and log⁡(y)=b\log (y)=b, then x=10ax = 10^a and y=10by = 10^b, and

x⋅y=10a⋅10b=10a+bx \cdot y = 10^a \cdot 10^b = 10^{a+b}

so that

log⁡(xy)=a+b\log(xy) = a+b

Examples​

Example
log⁡(100)=2log⁡(1000)=3\begin{aligned} \log(100) &= 2 \\ \log(1000) &= 3 \end{aligned}
Example

If

log⁡(2)≈0.3\log(2) \approx 0.3

then

10y=210^y=2

Note

210=1024≈1000=1032^{10}=1024 \approx 1000 = 10^3

therefore

2≈103/102 \approx 10^{3/10}

so

log⁡(2)≈0.3\log (2) \approx 0.3

The Natural Logarithm​

A logarithm with ee as a base is referred to as the natural logarithm and is denoted as ln⁡\ln:

y=ln⁡(x)y=\ln(x)

if

x=ey=exp⁡(y)x=e^y=\exp(y)

Note that ln⁡\ln is the inverse of exp⁡\exp.

Fig. 14

Figure: The curve depicts the function y=ln⁡(x)y=\ln(x) and shows that ln⁡\ln is the inverse of exp⁡\exp. +Note that ln⁡(1)=0\ln(1)=0 and when y=0y=0 then e0=1e^0=1.

Properties of Logarithm(s)​

Logarithms transform multiplicative models into additive models, i.e.

ln⁡(a⋅b)=ln⁡a+ln⁡b\ln(a\cdot b) = \ln a + \ln b

Details​

This implies that any statistical model, which is multiplicative becomes additive on a log scale, e.g.

y=a⋅wb⋅xcy = a \cdot w^b \cdot x^c

ln⁡y=(ln⁡a)+ln⁡(wb)+ln⁡(xc)\ln y = (\ln a) + \ln (w^b) + \ln (x^c)

Next, note that

ln⁡(x2)=ln⁡(x⋅x)=ln⁡x+ln⁡x=2⋅ln⁡x\begin{aligned} \ln (x^2) &= \ln (x \cdot x) \\ &= \ln x + \ln x \\ &= 2 \cdot \ln x \end{aligned}

and similarly ln⁡(xn)=n⋅ln⁡x\ln (x^n) = n \cdot \ln x for any integer nn.

In general ln⁡(xc)=c⋅ln⁡x\ln (x^c) = c \cdot \ln x for any real number c (for x>0x>0).

Thus the multiplicative model (from above)

y=a⋅wb⋅xcy=a \cdot w^b \cdot x^c

becomes

y=(ln⁡a)+b⋅ln⁡w+c⋅ln⁡xy= (\ln a) + b \cdot \ln w + c \cdot \ln x

which is a linear model with parameters (ln⁡a)(\ln a), bb and cc.

In addition, the log-transform is often variance-stabilizing.

The Exponential Function and the Logarithm​

The exponential function and the logarithms are inverses of each other

x=ey↔y=ln⁡xx = e^y \leftrightarrow y = \ln{x}

Details​

Note

Note the properties:

ln⁡(x⋅y)=ln⁡(x)+ln⁡(y)\ln (x \cdot y) = \ln (x) + \ln (y)

and

ea⋅eb=ea+be^a \cdot e^b = e^{a+b}

Examples​

Example

Solve the equation

10e1/3x+3=2410e^{1/3x} + 3 = 24

for xx.

First, get the 33 out of the way:

10e1/3x=2110e^{1/3x} = 21

Then the 1010:

e1/3x=2.1e^{1/3x} = 2.1

Next, we can take the natural log of 2.1. +Since ln⁡\ln is an inverse function of ee this would result in

13x=ln⁡(2.1)\displaystyle\frac{1}{3}x = \ln(2.1)

This yields

x=ln⁡(2.1)⋅3x = \ln(2.1) \cdot 3

which is

≈2.23\approx 2.23

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Miscellanea

Simple Probabilities In R​

R has functions to compute probabilities based on most common distributions

If XX is a random variable with a known distribution, then R can typically compute values of the cumulative distribution function or:

F(x)=P[X≤x]F(x)=P[X \leq x]

Examples​

Example

If X∼Bin(n,p)X \sim Bin(n,p) has binomial distribution, i.e.

P(X=x)=(nx)px(1−p)n−x,P(X = x) = \displaystyle{n \choose x}p^x(1-p)^{n-x},

then cumulative probabilities can be computed with pbinom\verb|pbinom|, e.g.

pbinom(5,10,0.5)

gives

P[X≤5]=0.623P[X \leq 5] = 0.623

where

X∼Bin(n=10,p=12)X \sim Bin(n=10,p= \displaystyle\frac{1}{2})

This can also be computed by hand. +Here we have n=10n=10, p=1/2p=1/2 and the probability P[X≤5]P[X \leq 5] is obtained by adding up the individual probabilities, P[X=0]+P[X=1]+P[X=2]+P[X=3]+P[X=4]+P[X=5]P[X =0]+P[X =1]+P[X =2]+P[X =3]+P[X =4]+P[X =5]

P[X≤5]=∑x=05(10x)12x1210−xP[X \leq 5] =\displaystyle\sum_{x=0}^5 \displaystyle{10\choose x} \displaystyle\frac{1}{2}^x\displaystyle\frac{1}{2}^{10-x}

This becomes

P[X≤5]=(100)1201210−0+(101)1211210−1+(101)1221210−2+(103)1231210−3+(104)1241210−4+(105)1251210−5P[X \leq 5] = \displaystyle{10 \choose 0} \displaystyle\frac{1}{2}^0\displaystyle\frac{1}{2}^{10-0} + \displaystyle{10 \choose 1} \displaystyle\frac{1}{2}^1\displaystyle\frac{1}{2}^{10-1} + \displaystyle{10 \choose 1} \displaystyle\frac{1}{2}^2\displaystyle\frac{1}{2}^{10-2} + \displaystyle{10 \choose 3} \displaystyle\frac{1}{2}^3\displaystyle\frac{1}{2}^{10-3} + \displaystyle{10 \choose 4} \displaystyle\frac{1}{2}^4\displaystyle\frac{1}{2}^{10-4} + \displaystyle{10 \choose 5} \displaystyle\frac{1}{2}^5\displaystyle\frac{1}{2}^{10-5}

or

P[X≤5]=(100)1210+(101)1210+(101)1210+(103)1210+(104)1210+(105)1210=1210(1+10+45+… )P[X \leq 5] = \displaystyle{10 \choose 0} \displaystyle\frac{1}{2}^{10} + \displaystyle{10 \choose 1} \displaystyle\frac{1}{2}^{10} + \displaystyle{10 \choose 1} \displaystyle\frac{1}{2}^{10} + \displaystyle{10 \choose 3} \displaystyle\frac{1}{2}^{10} + \displaystyle{10 \choose 4} \displaystyle\frac{1}{2}^{10} + \displaystyle{10 \choose 5} \displaystyle\frac{1}{2}^{10}=\displaystyle\frac{1}{2}^{10} \left(1+10+45+\dots \right)

Furthermore,

> pbinom(10,10,0.5)
[1] 1

and

> pbinom(0,10,0.5)
[1] 0.0009765625

It is sometimes of interest to compute P[X=x]P[X=x] in this case, and this is given by the dbinom function, e.g.

> dbinom(1,10,0.5)
[1] 0.009765625

or 101024\displaystyle\frac{10}{1024}

Example

Suppose XX has a uniform distribution between 0 and 1, i.e. X∼Unf(0,1)X \sim Unf(0,1). +Then the punifpunif function will return probabilities of the form

P[X≤x]=∫−∞xf(t)dt=∫0xf(t)dtP[X \leq x]= \int_{-\infty}^{x} f(t)dt= \int_{0}^{x} f(t)dt

where f(t)=1f(t)=1 if 0≤t≤10 \leq t \leq 1 and f(t)=0f(t)=0. +For example:

> punif(0.75)
[1] 0.75

To obtain P[a≤X≤b],P[a \leq X \leq b], we use punifpunif twice, e.g.

> punif(0.75)-punif(0.25)
[1] 0.5

Computing Normal Probabilities In R​

To compute probabilities X∼N(μ,σ2)X\sim N(\mu,\sigma^2) is usually transformed, since we know that

Z:=X−μσ∼(0,1)Z:=\displaystyle\frac{X-\mu}{\sigma} \sim(0,1)

The probabilities can then be computed for either XX or ZZ with the pnorm function in R.

Details​

Suppose XX has a normal distribution with mean μ\mu and variance

X∼N(μ,σ2)X\sim N(\mu,\sigma^2)

then to compute probabilities, XX is usually transformed, since we know that

Z=X−μσ∼(0,1)Z=\displaystyle\frac{X-\mu}{\sigma} \sim(0,1)

and the probabilities can be computed for either XX or ZZ with the pnorm function.

Examples​

Example

If Z∼N(0,1)Z \sim N(0,1) then we can e.g. obtain P[Z≤1.96]P[Z\leq1.96] with

> pnorm(1.96)
[1] 0.9750021

> pnorm(0)
[1] 0.5

> pnorm(1.96)-pnorm(1.96)
[1] 0

> pnorm(1.96)-pnorm(-1.96)
[1] 0.9500042

The last one gives the area between -1.96 and 1.96.

Example

If X∼N(42,32)X \sim N(42,3^2) then we can compute probabilities either by transforming

P[X≤x]=P[X−μσ≤x−μσ]=P[Z≤x−μσ]\begin{aligned} P[X\leq x] &= P\left[\displaystyle\frac{X-\mu}{\sigma} \leq \displaystyle\frac{x-\mu}{\sigma}\right] \\ &= P\left[Z \leq \displaystyle\frac{x-\mu}{\sigma}\right] \end{aligned}

and calling pnorm with the computed value z=x−μσz=\displaystyle\frac{x-\mu}{\sigma}, or call pnorm with xx and specify μ\mu and σ\sigma.

To compute P[X≤48]P[X\leq 48], either set z=(48−42)/3=2z=(48-42)/3=2 and obtain

> pnorm(2)
[1] 0.9772499

or specify μ\mu and σ\sigma

> pnorm(42,42,3)
[1] 0.5

Introduction to Hypothesis Testing​

Details​

If we have a random sample x1,…,xnx_1, \ldots, x_n from a normal distribution, then we consider them to be outcomes of independent random variables X1,…,XnX_1, \ldots, X_n where Xi∼N(μ,σ2)X_i \sim N(\mu, \sigma^2). +Typically, μ\mu and σ2\sigma^2 are unknown but assume for now that σ2\sigma^2 is known

Consider the hypothesis

H0:μ=μ0 vs. H1:μ>μ0H_0: \mu = \mu_0 \text{ vs. } H_1: \mu > \mu_0

where

μ0\mu_0

is a specified number.

Under the assumption of independence, the sample mean

x‾=1n∑i=1nxi\overline{x} = \displaystyle\frac{1}{n}\displaystyle\sum^n_{i=1}x_i

is also an observation from a normal distribution, with mean μ\mu but a smaller variance.Specifically, x‾\overline{x} is the outcome of

X‾=1n∑i=1nXi\overline{X} = \displaystyle\frac{1}{n}\displaystyle\sum^n_{i=1}X_i

and

X∼N(μ,σ2n)X \sim N(\mu, \displaystyle\frac{ \sigma^2}{n})

so the standard deviation of XX is σn\displaystyle\frac{\sigma}{\sqrt{n}}, so the appropriate error measure for x‾\overline{x} is σn\displaystyle\frac{\sigma}{\sqrt{n}}, when σ\sigma is unknown.

If H0H_0 is true, then

z:=x‾−μ0σ/nz:= \displaystyle\frac{\overline{x}-\mu_0}{\sigma / \sqrt{n}}

is an observation from an n∼N(0,1)n \sim N (0,1) distribution, i.e. an outcome of

Z=X‾−μ0σ/nZ= \displaystyle\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}

where Z∼N(0,1)Z \sim N(0,1) when H0H_0 is correct. +It follows that e.g. P[∣Z∣>1.96]=0.05P[\vert Z \vert > 1.96] = 0.05 and if we observe ∣Z∣>1.96\vert Z \vert > 1.96 then we reject the null hypothesis.

Note that the value z∗=1.96z^\ast = 1.96 is a quantile of the normal distribution and we can obtain other quantiles with the pnorm function, e.g. pnorm gives 1.961.96.

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Principles of Programming

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Principles of Programming

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Principles of Programming

Modularity​

Modularity involves designing a system that is divided into a set of functional units (named modules) that can be composed into a larger application. +Any programming project should be split into logical module pieces of code which are combined into a complete program.

Details​

Typically input, initialization, analysis, and output commands are grouped into separate parts.

Examples​

Example

Input:

dat <- read.table("http://notendur.hi.is/~gunnar/kennsla/alsm/data/set115.dat", header=T)
cols <- c("le", "osl")

Analysis:

Mn <- mean(dat[, cols[1]])

Output:

print(Mn)

Modularity and Functions​

In many cases groups of commands can be collected together into a function.

Details​

Typically a project has several such functions.

Examples​

Example

Suppose you want to plot the weight vs. length for several data sets in http://hi.is/~gunnar/kennsla/alsm/data.

A function can then be set up with the file number as an argument:

plotwtle <- function (fnum) {
  fname <- paste( "http://hi.is/~gunnar/kennsla/alsm/data/set",fnum,".dat",sep="")
  cat("The URL B", fname,"\n")
  dat <- read.table(fname,header=T)
  ttl <- paste("Data from file number", fnum)
  plot(dat$le,dat$osl,main=ttl)
}

Now call this with

plotwtle(105)

Modularity and Files​

It is advisable to split larger projects into several manageable files.

Details​

Once a project reaches more than five lines of code, it should be stored in one or more separate files. +In order to combine these files a single "source" command file can be created

Typically function definitions are stored in separate files, so one may have several separate files like: input.r, function.r, analysis.r, output.r. +While developing the analysis, the data would only be read once with source("input.r"). +The goal of this practice is to end up with a set of files which are completely self-contained, so one can start with an empty R session and give only the commands like:

  • source("input.r")

  • source("function.r")

  • source("analysis.r")

  • source("output.r")

Examples​

Example

For a given project input.r, function.r, analysis.r and output.r files can be created as below.

  • input.r

    dat <- read.table("http://notendur.hi.is/~gunnar/kennsla/alsm/data/set115.dat", header=T)

  • functions.r:

    plotwtle <- function(fnum) {
      fname <- paste("http://notendur.hi.is/~gunnar/kennsla/alsm/data/set",fnum,".dat",sep="")
      cat("The URL is",fname,"\n")
      dat <- read.table(fname,header=T)
      ttl <- paste("My data set was",fnum)
      plot(dat$le,dat$osl,main=ttl,xlab="Length(cm)",ylab="Live weight (g)")
    }

  • output.r:

    source("functions.r")
    for(i in 101:150) {
      fnam<-paste("plot",i,".pdf",sep="")
      pdf(fnam) plotwtle(i)
      dev.off()
    }

These files can be executed with source commands as below:

  • source("input.r")

  • source("function.r")

  • source("output.r")

Structuring an R Project​

Details​

We already covered how to split code into different functions and linking them together with the help of one executable file that is "sourcing" the others. +However, when you undertake a larger project, there will be a lot of different data and files and it is very advisable to have a consistent structure throughout the project

A common project layout is to allocate all project files into a folder, something along the lines of:

/project /data /src /doc /figs (or /out)

Such a structure is quite normal in programming languages such as C, Matlab, and R.

Purpose of the different folders:

  • /data: Contains all important data to the project, which you will use. +This folder should be read-only! No function is allowed to write anything into this folder

  • /src: (abbreviation for source(-code)) Here you will store all the functions that you programmed. +You can decide to store the executable function here as well or, alternatively, have that one in the root project folder

  • /doc: Contains further documentation material about your project. +This could be, for example, readme files for other people who use your functions, or the paper you wrote about the project, or the LaTeX\LaTeX files while you're writing

  • /figs or /out: Here your functions are allowed to write and can produce the different results, like graphs, figures or anything else

Finally, a large programming project should at some stage be split into packages and stored on dedicated servers such as GitHub or CRAN.

Examples​

Example

Consider first the issue of maintaining the code itself. +It is common for R beginners to only work interactively within the command-line interface. +However, it is essential that the code be kept in one or more files

For large projects these will be several different files, each with its own purpose. +To run a complete analysis one would typically set up one file to run all the tasks by reading in data through analyses to outputs

For example, a file named run.r could contain the sequence of commands:

source("setup.r")
source("analysis.r")
source("plot.r")

Loops, for​

If a piece of code is to be run repeatedly, the for-loop is normally used.

Details​

If a piece of code is to be run repeatedly, the for-loop is normally used. +The R code form is:

for(index in sequence){ commands }

Examples​

Example

To add numbers we can use

tot <- 100
for(i in 1:100){ tot <- tot + i }
cat("the sum is ", tot, "\n")
Example

Define the plot function

plotwtle <- AS BEFORE

To plot several of these we can use a sequence:

plotwtle(101)
plotwtle(102)
...

or a loop

for (i in 101:150) {
  fname <- paste("plot", i, ".pdf", sep="")
  pdf(fname)
  plotwtle(i)
  dev.off()
}

The if and ifelse Commands​

The if statement is used to conditionally execute statements

The ifelse statement conditionally replaces elements of a structure.

Examples​

Example

If we want to compute xxx^x for xx-values in the range 00 through 55, we can use

xlist <- seq(0,5,0.01)
y <- NULL
for(x in xlist) {
  if (x==0) {
    y <- c(y,1)
  }
  else {
    y <- c(y,x**x)
  }
}
Example
x <- seq(0,5,0.01)
y <- ifelse(x==0,1,x^x)
Example
dat <- read.table("file")
dat <- ifelse(dat==0,0.01,dat)
Example
x <- ifelse(is.na(x),0,x)

Indenting​

Code should be properly indented!

Details​

fFunctions, for-loops, and if-statements should always be indented.

Comments​

All code should contain informative comments. +Comments are separated out from code using the pound symbol (#).

Examples​

Example
#################### #####SETUP DATA##### ####################

dat <- read.table(filename)
x <- log(dat$le) #log-transformation of length
y <- log(dat$wt) #log-transformation of weight

###################### #####THE ANALYSIS##### ######################
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Sequences and Series

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Sequences and Series

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Sequences and Series

Sequences​

A sequence is a string of indexed numbers a1,a2,a3,…a_1, a_2, a_3, \ldots. +We denote this sequence with (an)n≥1(a_n)_{n\geq1}.

Details​

In a sequence the same number can appear several times in different places.

Examples​

Example

(1n)n≥1 is the sequence 1,12,13,14,…(\displaystyle\frac{1}{n})_{n\geq1} \text{ is the sequence } 1, \displaystyle\frac{1}{2}, \displaystyle\frac{1}{3}, \displaystyle\frac{1}{4}, \ldots

Example

(n)n≥1 is the sequence 1,2,3,4,5,…(n)_{n\geq1} \text{ is the sequence } 1, 2, 3, 4, 5, \ldots

Example

(2nn)n≥1 is the sequence 2,8,24,64,…(2^nn)_{n\geq1} \text{ is the sequence } 2, 8, 24, 64, \ldots

Convergent Sequences​

A sequence ana_n is said to converge to the number bb if for every ε>0\varepsilon >0 we can find an N∈NN\in \mathbb{N} such that ∣an−b∣<ε|a_n-b| < \varepsilon for all n≥Nn \geq N. +We denote this with lim⁡n→∞an=b\lim_{n\to\infty}a_n=b or an→ba_n\to b, as n→∞n\to\infty.

Details​

A sequence ana_n is said to converge to the number bb if for every ε>0\varepsilon >0 we can find an N∈NN\in \mathbb{N} such that ∣an−b∣<ε|a_n-b| < \varepsilon for all n≥Nn \geq N. +We denote this with lim⁡n→∞an=b\lim_{n\to\infty}a_n=b or an→ba_n\to b, as n→∞n\to\infty.

If xx is a number, then

(1+xn)n→ex as n→∞(1 + \displaystyle\frac{x}{n})^n \rightarrow e^x \text{ as } n\rightarrow\infty

Examples​

Example

The sequence (1n)n≥∞(\displaystyle\frac{1}{n})_{n\geq\infty} converges to 00 as n→∞n\to\infty.

Example

If x is a number, then

(1+xn)n→ex as n→∞(1 + \displaystyle\frac{x}{n})^n \rightarrow e^x \text{ as } n\rightarrow\infty

Infinite Sums (series)​

We are interested in, whether infinite sums of sequences can be defined.

Details​

Consider a sequence of numbers, (an)n→∞(a_n)_{n\to\infty}.

Now define another sequence (sn)n→∞,(s_n)_{n\to\infty}, where

sn=∑k=1naks_n=\displaystyle\sum_{k=1}^na_k

If (sn)n→∞(s_n)_{n\to\infty} is convergent to S=lim⁡n→∞sn,S=\lim_{n\to\infty}s_n, then we write

S=∑n=1∞anS=\displaystyle\sum_{n=1}^{\infty}a_n

Examples​

Example

If

ak=xk,qquadk=0,1,…a_k = x^k, qquad k=0,1,\dots

then

sn=∑k=0nxk=x0+x1+… .+xns_n=\displaystyle\sum_{k=0}^{n}x^k=x^0+x^1+\dots.+x^n

Note also that

xsn=x(x0+x1+… .+xn)=x+x2+⋯+xn+1xs_n=x(x^0+x^1+\dots.+x^n)= x + x^2 + \dots + x^{n+1}

We have

sn=1+x+x2+⋯+xns_n = 1 + x + x^2 + \dots + x^n

xsn=x+x2+⋯+xn+xn+1xs_n = x + x^2 + \dots +x^n + x^{n+1}

sn–xsn=1−xn+1s_n – xs_n = 1 - x^{n+1}

i.e.

sn(1−x)=1−xn+1s_n(1-x) = 1-x^{n+1}

and we have

sn=1−xn+11−xs_n =\displaystyle\frac{1-x^{n+1}}{1-x}

if x≠1x\neq1.

If 0<x<10< x<1 then xn+1→0x^{n+1}\to 0 as n→∞n\to\infty and we obtain sn→11−xs_n\to\displaystyle\frac{1}{1-x} so

∑n=0∞xn=11−x\displaystyle\sum_{n=0}^{\infty}x^n=\displaystyle\frac{1}{1-x}

The Exponential Function and the Poisson Distribution​

The exponential function can be written as a series (infinite sum):

ex=∑n=0∞xnn!e^x=\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac{x^n}{n!}

The Poisson distribution is defined by the probabilities

p(x)=e−λλxx! for x=0, 1, 2, …p(x)=e^{-\lambda}\displaystyle\frac{\lambda^x}{x!}\textrm{ for } x=0,\ 1,\ 2,\ \ldots

Details​

The exponential function can be written as a series (infinite sum):

ex=∑n=0∞xnn!e^x=\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac{x^n}{n!}

Knowing this we can see why the Poisson probabilities

p(x)=e−λλxx!p(x)=e^{-\lambda}\displaystyle\frac{\lambda^x}{x!}

add to one:

∑x=0∞p(x)=∑x=0∞e−λλxx!=e−λ∑x=0∞λxx!=e−λeλ=1\displaystyle\sum_{x=0}^{\infty}p(x)=\displaystyle\sum_{x=0}^{\infty}e^{-\lambda}\displaystyle\frac{\lambda^x}{x!}=e^{-\lambda}\displaystyle\sum_{x=0}^{\infty}\displaystyle\frac{\lambda^x}{x!}=e^{-\lambda}e^{\lambda}=1

Relation to Expected Values​

The expected value for the Poisson is given by

∑x=0∞xp(x)=∑x=0∞xe−λλxx!=λ\begin{aligned} \displaystyle\sum_{x=0}^\infty x p(x) &= \displaystyle\sum_{x=0}^\infty x e^{-\lambda} \displaystyle\frac{\lambda^x}{x!} \\ &= \lambda \end{aligned}

Details​

The expected value for the Poisson is given by

∑x=0∞xp(x)=∑x=0∞xe−λλxx!=e−λ∑x=1∞xλxx!=e−λ∑x=1∞λx(x−1)!=e−λλ∑x=1∞λ(x−1)(x−1)!=e−λλ∑x=0∞λxx!=e−λλeλ=λ\begin{aligned} \displaystyle\sum_{x=0}^\infty x p(x) &= \displaystyle\sum_{x=0}^\infty x e^{-\lambda} \displaystyle\frac{\lambda^x}{x!} \\ &= e^{-\lambda} \displaystyle\sum_{x=1}^\infty \displaystyle\frac{x\lambda^x}{x!} \\ &= e^{-\lambda} \displaystyle\sum_{x=1}^\infty \displaystyle\frac{\lambda^x}{(x-1)!} \\ &= e^{-\lambda} \lambda \displaystyle\sum_{x=1}^\infty \displaystyle\frac{\lambda^{(x-1)}}{(x-1)!} \\ &= e^{-\lambda} \lambda \displaystyle\sum_{x=0}^\infty \displaystyle\frac{\lambda^{x}}{x!} \\ &= e^{-\lambda} \lambda e^{\lambda} \\ &= \lambda \end{aligned}
+ + + + \ No newline at end of file diff --git a/Functions/Slopes of Lines and Curves/index.html b/Functions/Slopes of Lines and Curves/index.html index f0f2093..b0edec7 100644 --- a/Functions/Slopes of Lines and Curves/index.html +++ b/Functions/Slopes of Lines and Curves/index.html @@ -1,16 +1,16 @@ - + -Slopes of Lines and Curves | Computing and Calculus for Applied Statistics - - +Slopes of Lines and Curves | Computing and Calculus for Applied Statistics + +
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Slopes of Lines and Curves

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Slopes of Lines and Curves

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Slopes of Lines and Curves

The Slope of a Line​

Linear functions produce straight-line graphs. +In general, a straight line follows the following equation:

y=a+bxy = a + bx

where aa and bb are fixed numbers.

The line on the graph is the set of points:

[(x,y):x,y∈R,y=a+bx]\left[ (x,y): x,y \in \mathbb{R}, y = a+bx \right]

Fig. 22

Details​

The slope of a straight line represents the change in the yy coordinate corresponding to a unit change in the xx coordinate.

Segment Slopes​

Let's assume we have a more general function y=f(x)y = f(x). +To find the slope of a line segment, consider two xx -coordinates (x0x_0 and x1x_1), and look at the slope between (x0,f(x0))(x_0, f(x_0)) and (x1,f(x1))(x_1, f(x_1)).

Fig. 23

Details​

Consider two points, (x0,y0)(x_0,y_0) and (x1,y1)(x_1,y_1). +The slope of the straight line that goes through these points is

y1−y0x1−x0\displaystyle\frac {y_1 - y_0} {x_1 - x_0}

Thus, the slope of a line segment passing through the points (x0,f(x0))(x_0,f(x_0)) and (x1,f(x1))(x_1,f(x_1)), for some function, ff, is

f(x1)−f(x0)x1−x0\displaystyle\frac {f(x_1) - f(x_0)} {x_1 - x_0}

If we let x1=x0+hx_1 = x_0 + h then the slope of the segment is

f(x0+h)−f(x0)h\displaystyle\frac {f(x_0+h) - f(x_0)} {h}

The Slope of y=x2y=x^2​

Consider the task of computing the slope of the function y=x2y=x^2 at a given point.

Fig. 24

Examples​

Consider the function y=f(x)=x2y = f(x) = x^2. +In order to find the slope at a given point (x0)(x_0 ), we look at

y=f(x0+h)−f(x0)hy = \displaystyle\frac{f (x_0 +h) - f(x_0)} {h}

for small values of hh.

For this particular function, f(x)=x2f (x) = x^2, and hence

f(x0+h)=(x0+h)2=x2+2hx0+h2f (x_0 +h) = (x_0 +h) ^2 = x^2 + 2hx_0 + h^2

The slope of a line segment is therefore given by

f(x0+h)−f(x0)h=2hx0+h2h=2x0+h\displaystyle\frac{f (x_0 +h) - f(x_0)} {h}= \displaystyle\frac{2hx_0 + h^2} {h} = 2x_0 + h

As we make hh steadily smaller, the segment slope, 2x0+h2x_0 + h, tends towards 2x02x_0. +It follows that the slope, y′y', of the curve at a general point xx is given by y′=2xy' = 2x.

The Tangent to a Curve​

A tangent to a curve is a line that intersects the curve at exactly one point. +The slope of a tangent for the function y=f(x)y=f(x) at the point (x0,f(x0))(x_0,f(x_0)) is

lim⁡h→0f(x0+h)−f(x0)h\lim_{h\to0}\displaystyle\frac{f(x_0+h)-f(x_0)}{h}

Fig. 25

Details​

To find the slope of the tangent to a curve at a point, we look at the slope of a line segment between the points (x0,f(x0))(x_0,f(x_0)) and (x0+h,f(x0+h))(x_0+h,f(x_0+h)), which is

f(x0+h)−f(x0)h\displaystyle\frac{f(x_0+h)-f(x_0)}{h}

and then we take hh to be closer and closer to 00. +Thus the slope is

lim⁡h→0f(x0+h)−f(x0)h\lim_{h\to0}\displaystyle\frac{f(x_0+h)-f(x_0)}{h}

when this limit exists.

Examples​

Example

We wish to find the tangent line for the function f(x)=x2f(x)=x^2 at the point (1,1)(1,1). +First we need to find the slope of this tangent, it is given as

lim⁡h→0(1+h)2−12h=lim⁡h→02h+h2h=lim⁡h→0(2+h)=2\lim_{h\to0}\displaystyle\frac{(1+h)^2-1^2}{h}=\lim_{h\to0}\displaystyle\frac{2h+h^2}{h}=\lim_{h\to0}(2+h)=2

Then, since we know the tangent goes through the point (1,1)(1,1) the line is y=2x−1y=2x-1.

The Slope of a General Curve​

Fig. 26

Details​

Imagine a nonlinear function whose graph is a curve described by the equation y=f(x)y = f(x). +Here we want to find the slope of a line tangent to the curve at a specific point (x0)(x_0). +The slope of the line segment is given by the equation f(x0+h)−f(x0)h\displaystyle\frac{f (x_0 +h) - f(x_0)} {h}. +Reducing hh towards zero, gives the slope of this curve if it exists.

+ + + + \ No newline at end of file diff --git a/Functions/The Central Limit Theorem and Related Topics/index.html b/Functions/The Central Limit Theorem and Related Topics/index.html index a920485..fb96f53 100644 --- a/Functions/The Central Limit Theorem and Related Topics/index.html +++ b/Functions/The Central Limit Theorem and Related Topics/index.html @@ -1,16 +1,16 @@ - + -The Central Limit Theorem and Related Topics | Computing and Calculus for Applied Statistics - - +The Central Limit Theorem and Related Topics | Computing and Calculus for Applied Statistics + +
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The Central Limit Theorem and Related Topics

The Central Limit Theorem​

If measurements are obtained independently and come from a process with finite variance, then the distribution of their mean tends towards a Gaussian (normal) distribution as the sample size increases.

Fig. 30

Figure: The standard normal density

Details​

The Central Limit Theorem states that if X1,X2,…X_1, X_2, \ldots are independent and identically distributed random variables with mean μ\mu and (finite) variance σ2\sigma^2, then the distribution of Xˉn:=X1+⋯+Xnn\bar{X}_n:= \displaystyle\frac{X_1+\dots+X_n}{n} tends towards a normal distribution. +It follows that for a large enough sample size nn, the distribution random variable Xˉn\bar{X}_n can be approximated by N(μ,σ2/n)N(\mu,\sigma^2/n).

The standard normal distribution is given by the p.d.f.:

φ(z)=12πe−z22\varphi(z) = \displaystyle\frac{1}{\sqrt{2\pi}} e^{\displaystyle\frac{-z^2}{2}}

for z∈Rz\in \mathbb{R}

The standard normal distribution has an expected value of zero:

μ=∫zφ(z)dz=0\mu =\displaystyle\int z\varphi (z)dz =0

and a variance of:

σ2=∫(z−μ)2φ(z)dz=1\sigma^2 =\displaystyle\int ({z-\mu})^2 \varphi(z)dz=1

If a random variable ZZ has the standard normal (or Gaussian) distribution, we write Z∼N(0,1)Z\sim N(0,1).

If we define a new random variable, YY, by writing Y=σZ+μY=\sigma Z + \mu, then YY has an expected value of μ\mu, a variance of σ2\sigma^2 and a density (p.d.f.) given by the formula:

f(y)=12πσ e−(y−μ)22σ2f(y) = \displaystyle\frac{1}{\sqrt{2\pi}\sigma} \ e^{\displaystyle\frac{-(y-\mu)^2}{2\sigma^2}}

This is general normal (or Gaussian) density, with mean μ\mu and variance σ2\sigma^2.

The Central Limit Theorem states that if you take the mean of several independent random variables, the distribution of that mean will look more and more like a Gaussian distribution (if the variance of the original random variables is finite).

More precisely, the cumulative distribution function of:

Xˉn−μσ/n\displaystyle\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}}

converges to Φ\Phi, the N(0,1)N(0,1) cumulative distribution function.

Examples​

Example

If we collect measurements on waiting times, these are typically assumed to come from an exponential distribution with density

f(t)=λe−λt, for t>0f(t)=\lambda e^{-\lambda t},\textrm{ for } t>0

The Central Limit Theorem states that the mean of several such waiting times will tend to have a normal distribution.

Example

We are often interested in computing:

w=xˉ−μ0snw=\displaystyle\frac{\bar{x}-\mu_0}{\displaystyle\frac{s}{\sqrt{n}}}

which comes from a TT distribution (see below), if the xix_i are independent outcomes from a normal distribution.

However, if nn is large and σ2\sigma^2 is finite then ww values will look as though they came from a normal distribution. +This is in part a consequence of the Central Limit Theorem, but also of the fact that ss will become close to σ\sigma as nn increases.

Properties of the Binomial and Poisson Distributions​

The binomial distribution is really a sum of 00 and 11 values (counts of failures =0= 0 and successes =1=1). +So, a simple, single binomial outcome will correspond to coming from a normal distribution if the count is large enough.

Details​

Consider the binomial probabilities:

p(x)=(nx)px(1−p)n−xp(x)=\displaystyle\binom{n}{x}p^x(1-p)^{n-x}

for x=0,1,2,3,⋯ ,nx=0,1,2,3, \cdots,n where nn is a non-negative integer. +Suppose pp is a small positive number, specifically consider a sequence of decreasing pp -values, specified with pn=λnp_n= \displaystyle\frac{\lambda}{n} and consider the behavior of the probability as n→∞n \rightarrow \infty. +We obtain:

(nx)pnx(1−pn)n−x=n!x!(n−x!)(λn)x(1−λn)n−x=N(n−1)(n−2)⋯(n−x+1)x!λnx(1−λn)x(1−λn)n=N(n−1)(n−2)⋯(n−x+1)x!nxλx(1−λn)x(1−λn)n\begin{aligned} \displaystyle\binom{n}{x}p_n^x(1-p_n)^{n-x} &= \displaystyle\frac{n!}{x!(n-x!)} \left ( \displaystyle\frac{\lambda}{n} \right )^x \left ( 1-\displaystyle\frac{\lambda}{n} \right )^{n-x} \\ &= \displaystyle\frac{N(n-1)(n-2)\cdots (n-x+1)}{x!} \displaystyle\frac{\displaystyle\frac{\lambda}{n}^x}{\left ( 1-\displaystyle\frac{\lambda}{n} \right ) ^x} \left ( 1-\displaystyle\frac{\lambda}{n} \right )^n \\ &= \displaystyle\frac{N(n-1)(n-2)\cdots (n-x+1)}{x!n^x} \displaystyle\frac{\lambda^x}{\left ( 1-\displaystyle\frac{\lambda}{n} \right ) ^x} \left ( 1-\displaystyle\frac{\lambda}{n} \right )^n \end{aligned}
Note

Notice that N(n−1)(n−2)⋯(n−x+1)nx→1\displaystyle\frac{N(n-1)(n-2)\cdots (n-x+1)}{n^x}\to 1 as n→∞n\to\infty. +Also notice that (1−λn)x→1(1-\displaystyle\frac{\lambda}{n})^x\to 1 as n→∞n\to\infty. +Also

lim⁡n→∞(1−λn)=e−λ\lim_{n \to \infty} \bigg( 1-\displaystyle\frac{\lambda}{n} \bigg) = e^{- \lambda}

and it follows that

lim⁡n→∞(nx)pnx(1−pn)n−x=e−λλxx!,x=0,1,2,⋯ ,n\lim_{n \to \infty} \displaystyle\binom{n}{x}p_n^x(1-p_n)^{n-x} = \displaystyle\frac{e^{- \lambda} \lambda^x}{x!}, x= 0,1,2, \cdots, n

and hence the binomial probabilities may be approximated with the corresponding Poisson.

Examples​

Example

The mean of a binomial (n,p) variable is μ=n⋅p\mu=n\cdot p and the variance is σ2=np(1−p)\sigma^2=np(1-p).

The R command dbinom(q,n,p) calculates the probability of q successes in n trials assuming that the probability of a success is p in each trial (binomial distribution), and the R command pbinom(q,n,p) calculates the probability of obtaining q or fewer successes in n trials.

The normal approximation of this distribution can be calculated with pnorm(q,mu,sigma) which becomes pnorm(q,np,sqrt(np(1-p)). +Three numerical examples (note that pbinom and pnorm give similar values for large n):

> pbinom(3,10,0.2)
[1] 0.8791261

> pnorm(3,10*0.2,sqrt(10*0.2*(1-0.2)))
[1] 0.7854023
> pbinom(3,20,0.2)
[1] 0.4114489

> pnorm(3,20*0.2,sqrt(20*0.2*(1-0.2)))
[1] 0.2880751
> pbinom(30,200,0.2)
[1] 0.04302156

> pnorm(30,200*0.2,sqrt(200*0.2*(1-0.2)))
[1] 0.03854994
Example

We are often interested in computing w=xˉ−μs/nw=\displaystyle\frac{\bar{x}-\mu}{s/\sqrt{n}}, which has a TT distribution if the xix_i are independent outcomes from a normal distribution. +If nn is large and σ2\sigma^2 is finite, this will look as if it comes from a normal distribution.

The numerical examples below demonstrate how the TT distribution approaches the normal distribution.

> qnorm(0.7)
[1] 0.5244005 #This is the value which gives the cumulative probability of p=0.7 for a n~(0,1)

> qt(0.7,2)
[1] 0.6172134 #The value, which gives the cumulative probability of p=0.7 with n=2 for the t distribution.

> qt(0.7,5)
[1] 0.5594296

> qt(0.7,10)
[1] 0.541528

> qt(0.7,20)
[1] 0.5328628

> qt(0.7,100)
[1] 0.5260763

Monte Carlo Simulation​

If we know an underlying process we can simulate data from the process and evaluate the distribution of any quantity based on such data.

Fig. 31

Figure: A simulated set of tt-values based on data from an exponential distribution.

Examples​

Example

Suppose our measurements come from an exponential distribution and we want to compute

t=x‾−μs/nt = \displaystyle\frac{\overline x - \mu}{s / \sqrt{n}}

but we want to know the distribution of those when μ\mu is the true mean.

For instance, n=5n=5 and μ=1\mu = 1, we can simulate (repeatedly) x1,…,x5x_1, \ldots, x_5 and compute a tt -value for each. +The following R commands can be used for this:

library(MASS)
n <-5
mu <-1
lambda <-1
tvec <-NULL
for(sim in 1:10000) {
  x <-rexp(n,lambda)
  xbar <-mean(x)
  s <-sd(x)
  t <-(xbar-mu)/(s/sqrt(n))
  tvec <- c(tvec,t)
}

truehist(tvec) # truehist gives a better histogram

Show values at certain positions in the vector by uing:

> sort(tvec)[9750]
[1] 1.698656

> sort(tvec)[250]
[1] -6.775726
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Estimation, Estimates and Estimators

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Estimation, Estimates and Estimators

Ordinary Least Squares for a Single Mean​

If μ\mu is unknown and xi,…,xnx_i,\ldots,x_n are data, we can estimate μ\mu by finding

min⁡μ∑i=1n(xi−μ)2\min_{\mu} \displaystyle\sum_{i=1}^{n}(x_i-\mu)^2

In this case the resulting estimate is simply

μ=x‾\mu = \overline{x}

and can easily be derived by setting the derivative to zero.

Examples​

Example

Consider the numbers x1,…,x5x_1, \ldots, x_5 to be

13,7,4,16 and 913,7,4,16 \textrm{ and } 9

We can plot ∑(xi−μ)2\displaystyle\sum(x_i-\mu)^2 vs. μ\mu and find the minimum.

Maximum Likelihood Estimation​

If (Y1,…,Yn)′\left (Y_1, \ldots, Y_n\right )' is a random vector from a density fθf_{\theta} where θ\theta is an unknown parameter, and y\mathbf{y} is a vector of observations then we define the likelihood function to be

Ly(θ)=fθ(y)L_{\mathbf{y}}(\theta)=f_{\theta}(y)

Examples​

Example

If x1,…,xnx_1,\ldots,x_n are assumed to be observations of independent random variables with a normal distributions and mean of μ\mu and variance of σ2\sigma^2, then the joint density is

f(x1)⋅f(x2)⋅…⋅f(xn)f(x_1)\cdot f(x_2)\cdot\ldots\cdot f(x_n)

=12πσe−(x1−μ)22σ2⋅…⋅12πσe−(xn−μ)22σ2= \displaystyle\frac{1}{\sqrt{2\pi}\sigma}e^{-\displaystyle\frac{(x_1-\mu)^2}{2\sigma^2}} \cdot \ldots\cdot \displaystyle\frac{1}{\sqrt{2\pi}\sigma}e^{-\displaystyle\frac{(x_n-\mu)^2}{2\sigma^2}}

=Πi=1n12πσe−(xi−μ)22σ2=\Pi_{i=1}^n \displaystyle\frac{1}{\sqrt{2\pi}\sigma}e^{-\displaystyle\frac{(x_i-\mu)^2}{2\sigma^2}}

=1(2π)n/2σne−12σ2∑i=1N(xi−μ)2=\displaystyle\frac{1}{(2\pi)^{n/2}\sigma^n}e^{-\displaystyle\frac{1}{2\sigma^2}\displaystyle\sum_{i=1}^N(x_i-\mu)^2}

and if we assume σ2\sigma^2 is known then the likelihood function is

L(μ)=1(2π)n/2σne−12σ2Σi=1N(xi−μ)2L(\mu)=\displaystyle\frac{1}{(2\pi)^{n/2}\sigma^n}e^{-\displaystyle\frac{1}{2\sigma^2}\Sigma_{i=1}^N(x_i-\mu)^2}

Maximizing this is done by maximizing the log, i.e. finding the μ\mu for which:

ddμln⁡L(μ)=0,\displaystyle\frac{d}{d\mu}\ln L(\mu)=0,

which again results in the estimate

μ^=x‾\hat{\mu}=\overline{x}

Detail​

Definition

If (Y1,…,Yn)′\left (Y_1, \ldots, Y_n\right )' is a random vector from a density fθf_{\theta} where θ\theta is an unknown parameter, and y\mathbf{y} is a vector of observations then we define the likelihood function to be

Ly(θ)=fθ(y)L_{\mathbf{y}}(\theta)=f_{\theta}(y)

Ordinary Least Squares​

Consider the regression problem where we fit a line through (xi,yi)(x_i,y_i) pairs with x1,…,xnx_1, \ldots, x_n fixed numbers but where yiy_i is measured with error.

Fig. 36

Figure: Regression line through data pairs.

Details​

The ordinary least squares (OLS) estimates of the parameters α\alpha and β\beta in the model yi=α+βxi+ϵiy_i=\alpha + \beta x_i + \epsilon_i are obtained by minimizing the sum of squares

∑i(yi−(α+βxi))2\displaystyle\sum_i \left ( y_i -(\alpha +\beta x_i) \right )^2

a=y‾−bx‾b=∑i=1n(xi−x‾)(yi−y‾)∑i=1n(xi−x‾)2\begin{aligned} a &= \overline{y} - b\overline{x} \\ \\ b &= \displaystyle\frac{\displaystyle\sum^n_{i=1} (x_i-\overline{x})(y_i-\overline{y})}{\displaystyle\sum^n_{i=1} (x_i-\overline{x})^2} \end{aligned}

Random Variables and Outcomes​

Details​

Recall that X1,…,XnX_1, \ldots, X_n are random varibles (reflecting the population distribution) and x1,…,xnx_1, \ldots, x_n are numerical outcomes of these distributions. +We use upper case letters to denote random variables and lower case letters to denote outcome or data.

Examples​

Example

Let the mean of a population be zero and the σ=4\sigma=4. +Then draw three samples from this population with size, nn, either 44, 1616 or 6464. +The sample mean Xˉ\bar{X} will have a distribution with mean zero and standard deviation of σn\displaystyle\frac{\sigma}{\sqrt{n}} where n=4n= 4, 1616 or 6464.

Estimators and Estimates​

In OLS regression, note that the values of aa and bb:

a=y‾−bx‾a = \overline{y} - b \overline{x}

b=Σi=1n(xi−x‾)(yi−y‾)Σi=1n(xi−x‾)2b = \displaystyle\frac{\Sigma_{i=1}^{n} (x_i - \overline{x}) (y_i - \overline{y})}{\Sigma_{i=1}^{n} (x_i - \overline{x})^2}

are outcomes of random variables e.g. bb is the outcome of

β^=Σi=1n(xi−x‾)(Yi−Y‾)Σi=1n(xi−x‾)2\hat{\beta} = \displaystyle\frac{\Sigma_{i=1}^{n} (x_i - \overline{x}) (Y_i - \overline{Y})}{\Sigma_{i=1}^{n} (x_i - \overline{x})^2}

the estimator which has some distribution.

Fig. 37

Figure: Shows an example of the distribution of the estimator β^\hat{\beta}

Details​

The following R commands can be used to generate a distribution for the estimator β^\hat{\beta}

library(MASS)
nsim <- 1000
betahat <- NULL
for (i in 1:nsim) {
  n <- 20
  x <- seq(1:n) # Fixed x vector
  y <- 2 + 0.4*x + rnorm(n, 0, 1)
  xbar <- mean(x)
  ybar <- mean(y)
  b <- sum((x-xbar)*(y-ybar))/sum((x-xbar)^2)
  a <- ybar - b * xbar
  betahat <- c(betahat, b)
}
truehist(betahat)
+ + + + \ No newline at end of file diff --git a/Multivariate to Power/Multivariate Probability Distributions/index.html b/Multivariate to Power/Multivariate Probability Distributions/index.html index b5edcdd..72e7a9e 100644 --- a/Multivariate to Power/Multivariate Probability Distributions/index.html +++ b/Multivariate to Power/Multivariate Probability Distributions/index.html @@ -1,16 +1,16 @@ - + -Multivariate Probability Distributions | Computing and Calculus for Applied Statistics - - +Multivariate Probability Distributions | Computing and Calculus for Applied Statistics + +
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Multivariate Probability Distributions

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Multivariate Probability Distributions

Joint Probability Distribution​

If X1,…,XnX_1,\ldots, X_n are discrete random variables with P[X1=x1,X2=x2,…,Xn=xn]=p(x1,…,xn)P[X_1 = x_1, X_2 = x_2,\ldots, X_n = x_n] = p(x_1,\ldots, x_n), where x1,…,xnx_1, \ldots, x_n are numbers, then the function pp is the joint probability mass function (p.m.f.) for the random variables X1,…,XnX_1, \ldots, X_n.

For continuous random variables Y1,…,YnY_1, \ldots, Y_n, a function ff is called the joint probability density function if P[Y∈A]=∈∫…∫f(y1,…yn)dy1dy2⋯dynP [Y\in {A}] =\displaystyle\in\displaystyle\int\ldots\displaystyle\int f(y_1,\ldots y_n)dy_1dy_2 \cdots dy_n.

Details​

Definition

If X1,…,XnX_1, \ldots, X_n are discrete random variables with P[X1=x1,X2=x2,…,Xn=xn]=p(x1,…,xn)P[X_1 = x_1, X_2 = x_2,\ldots, X_n = x_n] = p(x_1,\ldots, x_n) where x1…xnx_1 \ldots x_n are numbers, then the function pp is the joint probability mass function (p.m.f.) for the random variables X1,…,XnX_1, \ldots, X_n.

Definition

For continuous random variables Y1,…,YnY_1, \ldots, Y_n, a function ff is called the joint probability density function if P[Y∈A]=∫∫…∫⏟Af(y1,…yn)dy1dy2⋯dynP [Y\in {A}] = \underbrace{\displaystyle\int\displaystyle\int\ldots\displaystyle\int}_{A} f(y_1,\ldots y_n)dy_1dy_2 \cdots dy_n.

Note

Note that if X1,…,XnX_1, \ldots, X_n are independent and identically distributed, each with p.m.f. pp, then p(x1,x2,…,xn)=q(x1)q(x2)…q(xn)p(x_1, x_2, \ldots, x_n) = q(x_1)q(x_2)\ldots q(x_n), i.e. P[X1=x1,X2=x2,…,Xn=xn]=P[X1=x1]P[X2=x2]…P[Xn=xn]P [X_1 = x_1, X_2 = x_2,\ldots, X_n= x_n] = P [X_1 = x_1] P[X_2 = x_2]\ldots P[X_n= x_n].

Note

Note also that if AA is a set of possible outcomes (A⊆Rn)(A \subseteq \mathbb{R}^n), then we have

P[X∈A]=∑(x1,…,xn)∈Ap(x1,…,xn)P[X \in {A}] = \displaystyle\sum_{(x_1,\ldots,x_n)\in A} p(x_1,\ldots, x_n)

Examples​

Example

An urn contains blue and red marbles, which are either light or heavy. +Let XX denote the color and YY the weight of a marble, chosen at random:

X ∖YLHTotalB5611R729TT12820\begin{array}{c c c c} \hline \\ X \setminus Y & \text{L} & \text{H} & \text{Total} \\ B & 5 & 6 & 11 \\ R & 7 & 2 & 9 \\ TT & 12 & 8 & 20 \\ \hline \end{array}

We have P[X="′b"′,Y="l"′]=520P[X="'b"', Y ="l"'] = \displaystyle\frac{5}{20}. +The joint p.m.f. is

X∖YLHTotalB5206201120R720220920Total12208201\begin{array}{c c c c} \hline \\ X \setminus Y & \text{L} & \text{H} & \text{Total} \\ \text{B} & \displaystyle\frac{5}{20} & \displaystyle\frac{6}{20} & \displaystyle\frac{11}{20} \\ \text{R} & \displaystyle\frac{7}{20} & \displaystyle\frac{2}{20} & \displaystyle\frac{9}{20} \\ \text{Total} & \displaystyle\frac{12}{20} & \displaystyle\frac{8}{20} & 1 \\ \hline \end{array}

The Random Sample​

A set of random variables X1,…,XnX_1, \ldots, X_n is a random sample if they are independent and identically distributed. +A set of numbers x1,…,xnx_1, \ldots, x_n are called a random sample if they can be viewed as an outcome of such random variables.

Fig. 32

Details​

Samples from populations can be obtained in a number of ways. +However, to draw valid conclusions about populations, the samples need to obtained randomly.

Definition

In random sampling, each item or element of the population has an equal and independent chance of being selected.

A set of random variables X1,…,XnX_1, \ldots, X_n is a random sample if they are independent and identically distributed.

Definition

If a set of numbers x1…xnx_1 \ldots x_n can be viewed as an outcome of random variables, these are called a random sample.

Examples​

Example

If X1,…,Xn∼Unf(0,1)X_1, \ldots, X_n \sim Unf(0,1), independent and identically distributed, i.e. X1X_1 and XnX_n are independent and each have a uniform distribution between 0 and 1. +Then they have a joint density which is the product of the densities of X1X_1 and XnX_n. +Given the data in the above figure and if x1,x2∈Rx_1, x_2 \in \mathbb{R}

f(x1,x2)=f1(x1)f2(x2)={1if 0≤x1,x2≤10elsewheref(x_1, x_2) = f_1(x_1) f_2(x_2) = \begin{cases} 1 & \text{if } 0 \leq x_1, x_2 \leq 1 \\ 0 & \text{elsewhere} \end{cases}
Example

Toss two dice independently, and let X1,X2X_1, X_2 denote the two (future) outcomes. +Then

P[X1=x1,X2=x2]={136if 1≤x1,x2≤60elsewhereP[X_1 = x_1, X_2 = x_2] = \begin{cases} \displaystyle\frac{1}{36} & \text{if } 1 \leq x_1, x_2 \leq 6 \\ 0 & \text{elsewhere} \end{cases}

is the joint p.m.f.

The Sum of Discrete Random Variables​

Details​

Suppose X and Y are discrete random values with a probability mass function p. +Let Z=X+Y. +Then

P(Z=z)=∑{(x,y):x+y=z}p(x,y)\begin{aligned} P(Z=z) & = &\displaystyle\sum_{\{ (x,y): x+y=z\}} p(x,y)\end{aligned}

Examples​

Example

(X,Y)=outcomes(X,Y) = \text{outcomes},

    [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    2    3    4    5    6    7
[2,]    3    4    5    6    7    8
[3,]    4    5    6    7    8    9
[4,]    5    6    7    8    9   10
[5,]    6    7    8    9   10   11
[6,]    7    8    9   10   11   12

P[X+Y=7]=636=16P[X+Y =7] =\displaystyle\frac{6}{36}=\displaystyle\frac{1}{6}

Because there are a total of 3636 equally likely outcomes and 77 occurs six times this means that P[X+Y=7]=16P[X + Y = 7] =\displaystyle\frac{1}{6}.

Also

P[X+Y=4]=336=112P[X+Y = 4] = \displaystyle\frac{3}{36} = \displaystyle\frac{1}{12}

The Sum of Two Continuous Random Variables​

If XX and YY are continuous random variables with joint p.d.f. ff and Z=X+YZ=X+Y, then we can find the density of ZZ by calculating the cumulative distribution function.

Fig. 33

Details​

If XX and YY are c.r.v. with joint p.d.f. ff and Z=X+YZ=X+Y, then we can find the density of ZZ by first finding the cumulative distribution function

P[Z≤z]=P[X+Y≤z]∫∫{(x,y):x+y≤z}f(x,y)dxdyP[Z \leq z]=P[X+Y \leq z]\displaystyle\int\displaystyle\int_{\{(x,y):x+y \leq z\}} f(x,y)dxdy

Examples​

Example

If X,Y∼Unf(0,1)X,Y \sim Unf(0,1), independent and Z=X+YZ=X+Y then

P[Z≤z]={0for z≤0z22for 0<z<11for z>21−(2−z)22for 1<z<2P[Z \leq z] = \begin{cases} 0 & \text{for } z \leq 0 \\ \displaystyle\frac{z^2}{2} & \text{for } 0 < z < 1 \\ 1 & \text{for } z > 2 \\ 1-\displaystyle\frac{(2-z)^2}{2} & \text{for } 1 < z < 2 \end{cases}

the density of zz becomes

g(z)={zfor 0<z≤12−zfor 1<z≤20for elsewhereg(z) = \begin{cases} z & \text{for } 0 < z \leq 1 \\ 2-z & \text{for } 1 < z \leq 2 \\ 0 & \text{for } \text{elsewhere} \end{cases}
Example

To approximate the distribution of Z=X+YZ=X+Y where X,Y∼Unf(0,1)X,Y \sim Unf(0,1) independent and identically distributed, we can use Monte Carlo simulation. +So, generate 10.000 pairs, set them up in a matrix and compute the sum.

Means and Variances of Linear Combinations of Independent Random Variables​

If XX and YY are random variables and a,b∈Ra,b\in\mathbb{R}, then

E[aX+bY]=aE[X]+bE[Y]E[aX+bY] = aE[X]+bE[Y]

Details​

If XX and YY are random variables, then

E[X+Y]=E[X]+E[Y]E[X+Y] = E[X]+E[Y]

i.e. the expected value of the sum is just the sum of the expected values. +The same applies to a finite sum, and more generally

E[∑i=1naiXi]=∑i=1naiE[Xi]E\left[\displaystyle\sum_{i=1}^{n} a_i X_i\right] = \displaystyle\sum_{i=1}^{n} a_i E[X_i]

when Xi,…,XnX_i,\dots,X_n are random variables and a1,…,ana_1,\dots,a_n are numbers (if the expectations exist).

If the random variables are independent, then the variance also add

Var[X+Y]=Var[X]+Var[Y]Var[X+Y] = Var[X] + Var[Y]

and

Var[∑i=1naiXi]=∑i=1nai2Var[Xi]Var\left[\displaystyle\sum_{i=1}^{n} a_i X_i\right] = \displaystyle\sum_{i=1}^{n} a_i^2 Var[X_i]

Examples​

Example

X,Y∼Unf(0,1)X,Y \sim Unf(0,1), independent and identically distributed, then

E[X+Y]=E[X]+E[Y]=∫01x⋅1dx∫01x⋅1dx=[12x2]01+[12x2]01=1E[X+Y]=E[X] + E[Y] =\displaystyle\int_0^1 x\cdot 1dx\displaystyle\int_0^1 x\cdot 1dx = \left[\displaystyle\frac{1}{2}x^2\right]_0^1+\left[\displaystyle\frac{1}{2}x^2\right]_0^1=1

Example

Let X,Y∼N(0,1)X,Y\sim N(0,1). +Then E[X2+Y2]=1+1=2E[X^2+Y^2] = 1+1=2.

Means and Variances of Linear Combinations of Measurements​

If x1,…,xnx_1,\dots,x_n and y1,…,yny_1,\dots,y_n are numbers, and we set

zi=xi+yiz_i=x_i + y_i

wi=axiw_i=ax_i

where a>0a>0, then

z‾=1n∑i=1nzi=x‾+y‾\overline{z} = \displaystyle\frac{1}{n} \displaystyle\sum_{i=1}^{n} z_i= \overline{x} + \overline{y}

w‾=ax‾\overline{w}= a\overline{x}

sw2=1n−1∑i=1n(wi−w‾)2s_w^2=\displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^{n}(w_i-\overline{w})^2

=1n−1∑i=1n(axi−ax‾)2= \displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^{n}(ax_i-a\overline{x})^2

=a2sx2= a^2s_x^2

and

sw=asxs_w=as_x

Examples​

Example

We set:

a < -3
x <- c(1:5)
y <- c(6:10)

Then:

z <- x+y
w <- a*x
n <-length(x)

Then z‾\overline{z} is:

> (sum(x)+sum(y))/n
[1] 11

> mean(z)
[1] 11

and w‾\overline{w} becomes:

> a*mean(x)
[1] 9

> mean(w)
[1] 9

and sw2s_w^2 equals:

> sum((w-mean(w))^2))/(n-1)
[1] 22.5

> sum((a*x - a*mean(x))^2)/(n-1)
[1] 22.5

> a^2*var(x)
[1] 22.5

and sws_w equals:

> a*sd(x)
[1] 4.743416

> sd(w)
[1] 4.743416

The Joint Density of Independent Normal Random Variables​

If Z1,Z2∼N(0,1)Z_1, Z_2 \sim N(0,1) are independent then they each have density

ϕ(x)=12πe−x22,x∈R\phi(x)=\displaystyle\frac{1}{\sqrt{2\pi}}e^{-\displaystyle\frac{x^2}{2}},x\in\mathbb{R}

and the joint density is the product f(z1,z2)=ϕ(z1)ϕ(z2)f(z_1,z_2)=\phi(z_1)\phi(z_2) or

f(z1,z2)=1(2π)2e−z122−z222f(z_1,z_2)=\displaystyle\frac{1}{(\sqrt{2\pi})^2} e^{\displaystyle\frac{-z_1^2}{2}-\displaystyle\frac{z_2^2}{2}}

Details​

If X∼N(μ1,σ12)X\sim N (\mu_1,\sigma_1^2) and Y∼N(μ2,σ22)Y\sim N(\mu_2, \sigma_2^2) are independent, then their densities are:

fX(x)=12πσ1e−(x−μ1)22σ12f_X (x) = \displaystyle\frac{1}{\sqrt{2\pi}\sigma_1} e^{\displaystyle\frac{-(x-\mu_1)^2}{2\sigma_1^2}}

and:

fY(y)=12πσ2e−(y−μ2)22σ22f_Y(y) = \displaystyle\frac{1}{\sqrt{2\pi}\sigma_2} e^{\displaystyle\frac{-(y-\mu_2)^2}{2\sigma_2^2}}

and the joint density becomes:

12πσ1σ2e−(x−μ1)22σ12−(y−μ2)22σ22\displaystyle\frac{1}{2\pi\sigma_1\sigma_2} e^{-\displaystyle\frac{(x-\mu_1)^2}{2\sigma_1^2}-\displaystyle\frac{(y-\mu_2)^2}{2\sigma_2^2}}

Now, suppose X1,…,Xn∼N(μ,σ2)X_1,\ldots,X_n\sim N(\mu,\sigma^2) are independent and identically distributed, then

f(x‾)=1(2π)n2σne−∑i=1n(xi−μ)2aσ2f(\underline{x})=\displaystyle\frac{1}{(2\pi)^{\displaystyle\frac{n}{2}}\sigma^n} e^{-\displaystyle\sum^{n}_{i=1} \displaystyle\frac{(x_i-\mu)^2}{a\sigma^2}}

is the multivariate normal density in the case of independent and identically distributed variables.

More General Multivariate Probability Density Functions​

Examples​

Example

Suppose XX and YY have the joint density

f(x,y)={2 0≤y≤x≤10 otherwisef(x,y) = \begin{cases} 2 & \text{ } 0\leq y \leq x \leq 1 \\ 0 & \text{ otherwise} \end{cases}

First notice that

∫R∫Rf(x,y)dxdy∫x=01∫y=0x2dydx=∫012xdx=1\displaystyle\int_{\mathbb{R}}\displaystyle\int_{\mathbb{R}}f(x,y)dxdy\displaystyle\int_{x=0}^{1}\displaystyle\int_{y=0}^x2dydx = \displaystyle\int_0^12xdx = 1

so ff is indeed a density function.

Now, to find the density of XX, we first find the c.d.f. of XX +First note that for a<0a<0 we have P[X≤a]=0P[X\leq a]=0, but, if a≥0a\geq 0, we obtain

FX(a)=P[X≤a]∫x0a∫y=0x2dydx=[x2]0a=a2F_X(a)=P[X\leq a]\displaystyle\int_{x_0}^a \displaystyle\int_{y=0}^x2dydx=[x^2]_0^a=a^2

The density of XX is therefore

fX(x)=dF(x)dx{2x 0≤x≤10 otherwisef_X(x) = \displaystyle\frac{dF(x)}{dx} \begin{cases} 2x & \text{ } 0\leq x \leq 1 \\ 0 & \text{ otherwise} \end{cases}

Handout​

If f:Rn→Rf: \mathbb{R}^n\rightarrow\mathbb{R} is such that P[X∈A]=∫A…∫f(x1,…,xn)dx1⋯dxnP[X \in A] =\displaystyle\int_A\ldots\displaystyle\int f(x_1,\ldots, x_n)dx_1\cdots dx_n and f(x)≥0f(x)\geq 0 for all x‾∈Rn\underline{x}\in \mathbb{R}^n, then ff is the joint density of

X=(X1⋮Xn)\mathbf{X}= \left( \begin{array}{ccc} X_1 \\ \vdots \\ X_n \end{array} \right)

If we have the joint density of some multidimensional random variable X=(X1,…,Xn)X=(X_1,\ldots,X_n) given in this manner, then we can find the individual density functions of the XiX_i 's by integrating the other variables.

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Power and Sample Sizes

The Power of a Test​

Suppose we have a method to test a null hypothesis against an alternative hypothesis. +The test would be \"controlled\" at some level α\alpha, i.e. P[Reject H0]≤αP[\text{Reject } H_0] \leq \alpha whenever H0H_0 is true

On the other hand, when H0H_0 is false one wants P[Reject H0]P[\text{Reject } H_0] to be as high as possible.

If the parameter to be tested is θ\theta and θ0\theta_0 is a value within H0H_0 and θ1\theta_1 is in H1H_1, then we want Pθ0[Reject H0]≤αP_{\theta_0}[\text{Reject } H_0] \leq \alpha and Pθ1[Reject H0]P_{\theta_1}[\text{Reject } H_0] as large as possible.

For a general θ\theta we write:

β(θ)=Pθ[Reject H0]\beta(\theta) = P_{\theta} [\text{Reject } H_0]

for the power of the test.

Details​

Do not use the phrase "accept".

The Power of Tests for Proportions​

Fig. 38

Examples​

Example

Suppose 7 students are involved in an experiment which is comprised of 7 trails and each trial consists of rolling a die 9 times.

Experiment 1: A student records a 00 if they toss an even number ( 2,4,62,4,6 ) (i.e. failure), and records a 11 if they toss an odd number ( 1,3,51,3,5 ) (i.e. success). +After tossing the die 9 times and recording a 00 or 11 the student tabulates the number of 11 's. +This process is repeated six more times

Data and outcomes:

xx equals the number of successes in nn trials equal Σi=1n\Sigma_{i=1}^n

Thus, xx equals number of odd numbers.

Question: Test whether p=P[Odd number]=12p=P[\text{Odd number}]=\displaystyle\frac{1}{2} that is H0:p=12H_0: p=\displaystyle\frac{1}{2} vs. H1:p≠12H_1: p\neq \displaystyle\frac{1}{2}

Solution: Now, xx is an outcome of X∼Bin(n,p)X \sim Bin(n,p). +We know from the CLT that

X−npnp(1−p)∼N(0,1)\displaystyle\frac{X-np}{\sqrt{np(1-p)}} \sim N(0,1)

write p0=12p_0=\displaystyle\frac{1}{2} so if H0:p=p0H_0:p=p_0 is true then

Z:=X−np0np0(1−p0)∼N(0,1)Z:=\displaystyle\frac{X-np_0}{\sqrt{np_0(1-p_0)}}\sim N(0,1)

so we reject H0H_0

if the observed value

z=x−np0np0(1−p0)z=\displaystyle\frac{x-np_0}{\sqrt{np_0(1-p_0)}}

is such that ∣z∣>z1−α2\left | z\right | >z_{1-\displaystyle\frac{\alpha}{2}}

Outcomes from 21 trials:

744346534553645435367\begin{matrix} 7 & 4 & 4 \\ 3 & 4 & 6 \\ 5 & 3 & 4 \\ 5 & 5 & 3 \\ 6 & 4 & 5 \\ 4 & 3 & 5 \\ 3 & 6 & 7 \end{matrix}

z=7−9⋅129⋅12⋅12=7−4.53⋅12=14−93=53<1.96z=\displaystyle\frac{7-9\cdot \displaystyle\frac{1}{2}}{\sqrt{9\cdot\displaystyle\frac{1}{2}\cdot \displaystyle\frac{1}{2}}}=\displaystyle\frac{7-4.5}{3\cdot\displaystyle\frac{1}{2}}=\displaystyle\frac{14-9}{3}=\displaystyle\frac{5}{3} < 1.96

So we do not reject the null hypothesis!

Note

We can rewrite the test statistics slightly

z=x−n2n⋅12⋅12=x−923⋅12=2x−93z=\displaystyle\frac{x-\displaystyle\frac{n}{2}}{\sqrt{n\cdot\displaystyle\frac{1}{2}\cdot\displaystyle\frac{1}{2}}}=\displaystyle\frac{x-\displaystyle\frac{9}{2}}{3\cdot\displaystyle\frac{1}{2}}=\displaystyle\frac{2x-9}{3}

Note

Note that we reject if 2x−93>1.96\displaystyle\frac{2x-9}{3}>1.96 > i.e. if > 2x>9+3⋅1.96≈9+6=152x>9+3\cdot 1.96 \approx 9+6=15 > x>7.5x>7.5 > (for x=8x=8 > or x=9x=9 > ) or 2x<9−3⋅1.962x<9-3 \cdot 1.96 >, x<1.5x<1.5 > (for x=0x=0 or x=1x=1 > ).

Example

Suppose 7 students are involved in an experiment which is comprised of 7 trails and each trial consists of rolling a die 9 times

Experiment 2: The procedure is the same as in experiment 11, but now the student records 00 for a 11 or 22 (failure) and a 11 for a 33, 44, 55, or 66 (success).

Data and outcomes:

xx equals the number of successes in nn trials equal Σi=1n\Sigma_{i=1}^n +Thus, xx equaks number of 1's.

Solution: Outcomes from 21 experiments:

543857573765788564257\begin{matrix} 5 & 4 & 3 \\ 8 & 5 & 7 \\ 5 & 7 & 3 \\ 7 & 6 & 5 \\ 7 & 8 & 8 \\ 5 & 6 & 4 \\ 2 & 5 & 7 \end{matrix}

This time our test is H0:p=23H_0:p=\displaystyle\frac{2}{3} vs H1:p=23H_1:p=\displaystyle\frac{2}{3}. +Note that we reject H0H_0 if 6x−4n9>1,96\displaystyle\frac{6x-4n}{9}>1,96 (for x=9x=9 ) or if 6x−4n9<−1,96\displaystyle\frac{6x-4n}{9}<-1,96 (for x=0,1,2,3x=0,1,2,3)

We reject H0H_0 in 3 out of 21 trials.

Example

Suppose 7 students are involved in an experiment which is comprised of 7 trails and each trial consists of rolling a die 9 times.

Experiment 3: Same as experiment 11 except 00 is recorded for 1,2,3,4,51,2,3,4,5 (failure) and a 11 is recorded for 66 (success)

Data and outcomes:

xx equals number of successes in nn trials equal Σi=1n\Sigma_{i=1}^n +Thus, xx equals number of 11 's.

Solution: Outcomes from 21 experiments

012121142111131112020\begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 1 \\ 1 & 4 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 2 \\ 0 & 2 & 0 \end{matrix}

With the same kind of calculations as above, we find that we reject the null hypothesis H0:p=16H_0:p=\displaystyle\frac{1}{6} in 14 out of 21 trials.

The Power of the One Sided zz -test for the mean​

The one sided zz -test for the mean (μ)(\mu) is based on a random sample where X1,…,Xn∼N(μ,σ2)X_1, \ldots,X_n \sim N(\mu, \sigma^2) are independent and σ2\sigma^2 is known. +The power of the test for an arbitrary μ\mu can be computed as:

β(μ)=1−Φ(μ0−μσn+z1−α)\beta(\mu) = 1 - \Phi \left ( \displaystyle\frac{\mu_0 - \mu }{\displaystyle\frac{\sigma} {\sqrt{n}}} + z_{1- \alpha} \right )

Details​

The one sided zz -test for the mean (μ)(\mu) is based on a random sample where X1,…,Xn∼N(μ,σ2)X_1, \ldots,X_n \sim N(\mu, \sigma^2) are independent and σ2\sigma^2 is known. +If the hypotheses are H0:μ=μ0H_0 : \mu = \mu_0 vs H1:μ>μ0H_1 : \mu > \mu_0, then we know that, if H0H_0 is true.

Z=Xˉ−μ0σn∼N(0,1)Z = \displaystyle\frac{\bar {X} - \mu_0}{\displaystyle\frac{\sigma} {\sqrt{n}}} \sim N (0,1)

Given data x1,…,xnx_1, \ldots, x_n, the zz-value is

z=xˉ−μ0σnz = \displaystyle\frac{\bar {x} - \mu_0}{\displaystyle\frac{\sigma} {\sqrt{n}}}

We reject H0H_0 if z>z1−αz > z_{1-\alpha}

The level of this test is:

Pμ0[Reject H0]=Pμ0[Xˉ−μ0σn>z1−α]P_{\mu_0} [\text{Reject } H_0]= P_{\mu_0}[\displaystyle\frac{\bar {X} - \mu_0}{\displaystyle\frac{\sigma} {\sqrt{n}}} > z_{1- \alpha}]

=P[z>z1−α]=α= P[z > z_{1- \alpha}] = {\alpha}

since Z∼N(0,1)Z \sim N(0,1) when μ0\mu_0 is the true value.

The power of the test for an arbitrary μ\mu can be computed as follows.

β(μ)=Pμ[Reject H0]=Pμ[Xˉ−μ0σn>z1−α]=Pμ[Xˉ>μ0+z1−ασn]=Pμ[Xˉ−μσn>μ0−μσn+z1−α]=P[Z>μ0−μσn+z1−α]\begin{aligned} \beta(\mu) &= P_{\mu} [\text{Reject } H_0] \\ &= P_{\mu}\left[\displaystyle\frac{\bar {X} - \mu_0}{\displaystyle\frac{\sigma} {\sqrt{n}}} > z_{1- \alpha}\right] \\ &= P_{\mu} \left[\bar {X}> \mu_0 + z_{1- \alpha}{\displaystyle\frac{\sigma} {\sqrt{n}}}\right] \\ &= P_{\mu} \left[\displaystyle\frac{\bar {X} - \mu}{\displaystyle\frac{\sigma} {\sqrt{n}}}> \displaystyle\frac{\mu_0 - \mu }{\displaystyle\frac{\sigma} {\sqrt{n}}}+ z_{1- \alpha}\right] \\ &= P\left[Z > \displaystyle\frac{\mu_0 - \mu}{\displaystyle\frac{\sigma} {\sqrt{n}}} + z_{1- \alpha}\right]\end{aligned}

We obtain:

β(μ)=1−Φ(μ0−μσn+z1−α)\beta(\mu) = 1 - \Phi \left ( \displaystyle\frac{\mu_0 - \mu }{\displaystyle\frac{\sigma} {\sqrt{n}}} + z_{1- \alpha} \right )

Examples​

Example

Suppose we know σ=2\sigma=2 and we will take a sample from n(μ,σ2)n\left ( \mu, \sigma^2\right) intending to test the hypothesis μ=3\mu=3 at level α=0.05\alpha=0.05. +We want to know the power against a one-tailed alternative when the true mean is actually μ=4\mu=4 when the sample size is n=25n=25.

We can set this up in R with:

alpha <- 0.05
n <- 25
sigma <- 2
mu0 <-3
mu <-4
zcrit <- qnorm(1-alpha)

Sticking the formula into R gives

> 1-pnorm((mu0-mu)/(sigma/sqrt(n))+zcrit)
[1] 0.803765

On the other hand, one can also use a simple simulation approach. +First, decide how many samples are to be simulated ( Nsim ). +Then, generate all of these samples, arrange them in a matrix and compute the mean of each sample. +The zz -value of each of these Nsim tests are then computed and a check is made whether it exceeds the critical point (1) or not (0).

Nsim <- 10000
m <- matrix(rnorm(Nsim*n,mu,sigma),ncol=n)
mn <- apply(m,1,mean)
z <- (mn-mu0)/(sigma/sqrt(n))
i<-ifelse(z>zcrit,1,0)

Compute the sum:

> sum(i/Nsim)
[1] 0.8081

Power and Sample Size for the One-sided zz -test for a single normal mean​

Suppose we want to test H0:μ=μ0H_0:\mu=\mu_0 vs H1:μ>μ0H_1:\mu>\mu_0. +We will reject H0H_0 if the observed value

z=x‾−μ0σ/nz=\displaystyle\frac{\overline{x}-\mu_0}{\sigma/\sqrt{n}}

is such that z>z1−αz>z_{1-\alpha}.

Details​

Suppose we want to test H0:μ=μ0H_0:\mu=\mu_0 vs H1:μ>μ0H_1:\mu>\mu_0. +So based on X1,…,Xn∼N(μ,σ2)X_1,\dots,X_n\sim N(\mu,\sigma ^2) independent and identically distributed with σ2\sigma^2 known we will reject H0H_0 if the observed value

z=x‾−μ0σ/nz=\displaystyle\frac{\overline{x}-\mu_0}{\sigma/\sqrt{n}}

is such that z>z1−αz>z_{1-\alpha}.

The power is given by:

β(μ)=1−Φ(μ−μ0σ/n+z1−α)\beta(\mu)=1-\Phi(\displaystyle\frac{\mu-\mu_0}{\sigma/\sqrt{n}}+z_{1-\alpha})

and describes the probability of rejecting H0H_0 when μ\mu is the correct value of the parameter. +Suppose we want to reject H0H_0 with a prespecified probability β1\beta_1, when μ1\mu_1 is the true value of μ\mu. +For this, we need to select the sample size so that

β(μ1)≥β1\beta(\mu_1) \geq \beta_1

i.e. find nn which satisfies

1−Φ(μ1−μ0σ/n+z1−α)≥β11-\Phi\left(\displaystyle\frac{\mu_1-\mu_0}{\sigma/\sqrt{n}}+z_{1-\alpha}\right)\geq \beta_1

Examples​

Example
mu0 <-10
sigma <- 2
mu1 <- 11
n <- 50
d <- (mu1-mu0)
power.t.test(n=n,delta=d,sd=sigma,sig.level=0.05,type="one.sample",alternative="one.sided",strict=TRUE)

     One-sample t test power calculation

              n = 50
          delta = 1
             sd = 2
      sig.level = 0.05
          power = 0.9672067
    alternative = one.sided

The Non Central T - Distribution​

Recall that if Z∼N(0,1)Z \sim N(0, 1) and U∼χ2vU \sim {\chi^2}_v are independent then

ZUv∼tv\displaystyle\frac{Z}{\sqrt{\displaystyle\frac{U}{v}}}\sim t_v

and it follows for a random sample X1,…,Xn∼N(μ,σ2)X_1, \ldots,X_n \sim N(\mu, \sigma^2) independent; that

Xˉ−μsn=Xˉ−μσn∑(Xi−Xˉ)2σ2n−1∼tn−1\displaystyle\frac{\bar {X} - \mu}{\displaystyle\frac{s} {\sqrt{n}}} = \displaystyle\frac{\displaystyle\frac{\bar {X} - \mu}{\displaystyle\frac{\sigma} {\sqrt{n}}}}{\sqrt{\displaystyle\frac{\displaystyle\sum ({X_i} -\bar {X})^2} { \displaystyle\frac {{\sigma}^2} {n-1}}}} \sim t_{n-1}

Details​

On the other hand, if W∼N(Δ,1)W \sim N (\Delta,1) and U∼χv2U \sim {\chi}^2_v are independent, then WUv\displaystyle\frac{W}{\sqrt{\displaystyle\frac{U}{v}}} has a non central TT distribution with vv degrees of freedom and non centrality parameter Δ\Delta. +This distribution arises, if X1,…,Xn∼N(μ,σ2)X_1, \ldots,X_n \sim N(\mu, \sigma^2) independent and we want to consider the distribution of:

Xˉ−μSn=Xˉ−μσn+μ−μ0σnSn=Z+μ−μ0σnUv\displaystyle\frac{\bar {X} - \mu}{\displaystyle\frac{S} {\sqrt{n}}} = \displaystyle\frac{\displaystyle\frac{\bar {X} - \mu}{\displaystyle\frac{\sigma} {\sqrt{n}}} + \displaystyle\frac{\mu - \mu_0 }{\displaystyle\frac{\sigma} {\sqrt{n}}}} {\displaystyle\frac{S}{\sqrt{n}}} = \displaystyle\frac {Z + \displaystyle\frac{\mu - \mu_0 }{\displaystyle\frac{\sigma} {\sqrt{n}}}}{\sqrt{\displaystyle\frac{U}{v}}}

Where μ≠μ0\mu \neq \mu_0 which is a non central t with non centrality parameters:

Δ=μ−μ0σn\Delta = \displaystyle\frac{\mu - \mu_0 }{\displaystyle\frac{\sigma} {\sqrt{n}}}

with n−1n-1 df. +Here v=n−1v = n-1 df since Z∼N(0,1)Z \sim N(0,1) and U∼χn−12U \sim {\chi}^2_{n-1} in this equation.

The Power of T-test for a Normal Mean (warning: Errors)​

Details​

WARNING: This is all wrong and needs to be rewritten.

Consider X1,…,Xn∼N(μ,σ2)X_1,\ldots,X_n \sim N (\mu, \sigma^2) independent and identically distributed where σ2\sigma^2 is unknown and we want to test H0:μ=μ0H_0:\mu = \mu_0 vs. H1:μ>μ0H_1: \mu > \mu_0. +We know that

T:=X‾−μs/n∼tn−1T:= \displaystyle\frac{\overline{X} - \mu}{s/\sqrt{n}} \sim t_{n-1}

and we will reject H0H_0 if the computed value

t:=x‾−μ0s/nt:= \displaystyle\frac{\overline{x} - \mu_0}{s/\sqrt{n}}

is such that

t>t∗=tn−1,1−αt>t^{\ast}=t_{n-1, 1-\alpha}

The power of this test is:

B(μ)=Pμ[Reject H0]=Pμ[x‾−μ0s/n>t∗]=Pμ[x‾−μ0>t∗⋅s/n]=Pμ[x‾−μs/n>t∗+μ0−μs/n]\begin{aligned} B(\mu) = P_{\mu}[\text{Reject } H_0] &= P_{\mu} \lbrack \displaystyle\frac{\overline{x} - \mu_0}{s/\sqrt{n}} > t^\ast \rbrack \\ &=P_{\mu} \lbrack \overline{x} - \mu_0 > t^\ast\cdot s/\sqrt{n} \rbrack \\ &=P_{\mu} \lbrack \displaystyle\frac{\overline{x} - \mu}{s/\sqrt{n}} > t^\ast+\displaystyle\frac{\mu_0-\mu}{s/\sqrt{n}} \rbrack \end{aligned}

Which is the probability that a tn−1,1−αt_{n-1,1-\alpha} -variable exceed t∗+μ0−μs/nt^{\ast}+\displaystyle\frac{\mu_0-\mu}{s/\sqrt{n}}.

WARNING: This is all wrong and needs to be rewritten (the s in the above line is a random variable so this make no sense at all).

Power and Sample Size for the One Sided T-test for a Mean​

Suppose we want to calculate the power of a one sided tt-test for a single mean (one sample), this can easily be done in R with the power.t.test command.

Details​

Δ=μ1−μ2\Delta = \mu_1-\mu_2 \ δ=μ1−μ2σ/n\delta = \displaystyle\frac{\mu_1-\mu_2}{\sigma/\sqrt{n}}

Examples​

Example

For a one sided power analysis we wish to test the following hypotheses:

For a one sample test: H0:μ=μ0H_0: \mu = \mu_0 vs. H1:μ>μ0H_1: \mu > \mu_0

For a two sample test: H0:μ1=μ2H_0: \mu_1 = \mu_2 vs. H1:μ1>μ2H_1: \mu_1 > \mu_2

In R, the power.t.test command is useful to calculate how many samples one needs to obtain a certain power of a test, but also to calculate the power when we have a given number of samples.

Example

How many samples do I need to get a power of 0.90.9?

> power.t.test(power=.95, delta=1.5,sd=2, type="one.sample", alternative="one.sided")

     One-sample t test power calculation

              n = 20.67702
          delta = 1.5
             sd = 2
      sig.level = 0.05
          power = 0.95
    alternative = one.sided

We would thus need a sample size of $n = 31.15 \approx 32$ samples to obtain a power of $0.9$ for our analysis.
Example

With a sample size of n=45n = 45, what will the power of my test be?

> power.t.test(n=45,delta=1.5,sd=2,sig.level=0.05,type="one.sample",alternative="one.sided")

     One-sample t test power calculation

              n = 45
          delta = 1.5
             sd = 2
      sig.level = 0.05
          power = 0.9995287
    alternative = one.side

This is done the same way for two samples only by changing the alternative to two.sample. +For two sided power analysis, one only needs to change the alternative to two.sided.

Example

If one is interested in doing a power analysis for an ANOVA test, this is done in a fairly similar way. +With a given sample size of n=20n=20:

> power.anova.test(groups=4, n=20, between.var=1, within.var=3)

     Balanced one-way analysis of variance power calculation

         groups = 4
              n = 20
    between.var = 1
     within.var = 3
      sig.level = 0.05
          power = 0.9679022

To calculate the sample size needed to obtain a power of 0.90 for a test:

> power.anova.test(groups=4, between.var=1, within.var=3, power=.9)

     Balanced one-way analysis of variance power calculation

         groups = 4
              n = 15.18834
    between.var = 1
     within.var = 3
      sig.level = 0.05
          power = 0.9

NOTE: n is number in each group

The Power of the 2-sided T-test​

A power analysis on a two-sided tt-test can be done in R using the power.t.test command.

Details​

For a one sample test, H0:μ=μ0H_0: \mu=\mu_0 vs. H1:μ≠μ0H_1:\mu\neq\mu_0, the power.t.test command is useful to provide information for determining the minimum sample size one needs to obtain a certain power of a test:

power.t.test(n=,delta=,sd=,sig.level=,power=,type=c("two.sample","one.sample","paired"),alternative=c("two.sided"))

where:

  • n is the sample size
  • dr is the effect size
  • s is the standard deviation
  • sig.level is the significance level
  • power is normally 0.8, 0.9 or 0.95
  • type is two sample, one sample or paired (the type selected depends on the research)
  • alternative is either one sided or two sided

Examples​

Example

How many samples do I need in my research to obtain a power of 0.8?

> power.t.test(delta=1.5,sd=2,sig.level=0.05,power=0.8,type=c("two.sample"),alternative=c("two.sided"))

     Two-sample t test power calculation

              n = 28.89962
          delta = 1.5
             sd = 2
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: n is number in *each* group

So, one needs 29 samples (n=29) to obtain a power level of 0.8 for this analysis.

The Power of the 2-sample One and Two-sided T-tests​

The power of a two sample, one-sided t-test can be computed as follows:

β(μ1μ2)=Pμ1μ2[Z+ΔU/(n+m−2)>t1−α,n+m−2∗]\beta_{(\mu_1\mu_2)} = P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right]

and the power of a two sample, two-sided t-test is given by:

β(μ1μ2)=Pμ1μ2[Z+ΔU/(n+m−2)>t1−α,n+m−2∗]+Pμ1μ2[Z+ΔU/(n+m−2)<−t1−α,n+m−2∗]\begin{aligned} \beta_{(\mu_1\mu_2)} &= P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right] \\ &+ P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} < -t^\ast_{1-\alpha,n+m-2} \right] \end{aligned}

where Δ=(μ1−μ2)σ1n+1m\Delta = \displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}} and U is the SSE.

Details​

Two Sample, one-sided t-test

Suppose data are gathered independently from two normal populations resulting in

X1,…,Xn∼N(μ1,σ2)X_1, \ldots, X_n \sim N(\mu_1, \sigma^2)

Y1,…,Ym∼N(μ2,σ2)Y_1, \ldots, Y_m \sim N(\mu_2, \sigma^2)

where all data are independent then

X‾−Y‾∼N(μ1−μ2,σ2n +σ2m)\overline{X}-\overline{Y} \sim N(\mu_1 - \mu_2, \displaystyle\frac{\sigma^2}{n}\ + \displaystyle\frac{\sigma^2}{m})

The null hypothesis in question is H0:μ1=μ2H_0: \mu_1 = \mu_2 versus the alternative H1:μ1>μ2H_1: \mu_1 > \mu_2. +If H0H_0 is true then the observed value

t=x‾−y‾s1n+1mt = \displaystyle\frac{\overline{x}-\overline{y}}{s\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}

comes from a TT distribution with n+m−2n+m-2 degrees of freedom and we reject H0H_0 if ∣t∣>t1−α,n+m−2∗\left|t \right|> t^\ast_{1-\alpha,n+m-2} \ The power of the test can be computed as follows:

β(μ1μ2)=Pμ1μ2[Reject H0]=Pμ1μ2[X‾−Y‾S1n+1m>t1−α,n+m−2∗]=Pμ1μ2[X‾−Y‾−(μ1−μ2)σ1n+1m+(μ1−μ2)σ1n+1mS/σ>t1−α,n+m−2∗]=Pμ1μ2[Z+(μ1−μ2)σ1n+1mS/(n+m−2)>t1−α,n+m−2∗]=Pμ1μ2[Z+ΔU/(n+m−2)>t1−α,n+m−2∗]\begin{aligned} \beta_{(\mu_1\mu_2)} &= P_{\mu_1\mu_2}\left[\text{Reject } H_0 \right] \\ &= P_{\mu_1\mu_2}\left[\displaystyle\frac{\overline{X}-\overline{Y}}{S\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}} > t^\ast_{1-\alpha,n+m-2} \right] \\ &= P_{\mu_1\mu_2}\left[\displaystyle\frac{\displaystyle\frac{\overline{X}-\overline{Y}-(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}+ \displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}}{S/\sigma} > t^\ast_{1-\alpha,n+m-2}\right] \\ &= P_{\mu_1\mu_2}\left[\displaystyle\frac{Z +\displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}} {S/\sqrt{(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right] \\ &= P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right] \end{aligned}

where Δ=(μ1−μ2)σ1n+1m\Delta = \displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}

and UU is the SSE of the samples which is divided by the appropriate degrees of freedom to give a χ2\chi^2 distribution.

This is the probability that a non-central t-variable exceeds t∗t^\ast.

Two Sample, Two-sided t-test

In this case the null hypothesis is defined as H0:μ1=μ2H_0: \mu_1 = \mu_2 versus alternative H1:μ1≠μ2H_1: \mu_1 \neq \mu_2.

The power of the test can be computed as follows:

β(μ1μ2)=Pμ1μ2[Reject H0]=Pμ1μ2[∣X‾−Y‾S1n+1m∣>t1−α,n+m−2∗]=Pμ1μ2[X‾−Y‾S1n+1m>t1−α,n+m−2∗]+Pμ1μ2[X‾−Y‾S1n+1m<−t1−α,n+m−2∗]=Pμ1μ2[X‾−Y‾−(μ1−μ2)σ1n+1m+(μ1−μ2)σ1n+1mS/(n+m−2)>t1−α,n+m−2∗]+Pμ1μ2[X‾−Y‾−(μ1−μ2)σ1n+1m+(μ1−μ2)σ1n+1mS/(n+m−2)<−t1−α,n+m−2∗]=Pμ1μ2[Z+ΔU/(n+m−2)>t1−α,n+m−2∗]+Pμ1μ2[Z+ΔU/(n+m−2)<−t1−α,n+m−2∗]\begin{aligned} \beta_{(\mu_1\mu_2)} &= P_{\mu_1\mu_2}\left[\text{Reject } H_0 \right] \\ &= P_{\mu_1\mu_2}\left[ \left|\displaystyle\frac{\overline{X}-\overline{Y}}{S\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}\right| > t^\ast_{1-\alpha,n+m-2} \right] \\ &= P_{\mu_1\mu_2}\left[ \displaystyle\frac{\overline{X}-\overline{Y}}{S\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}} > t^\ast_{1-\alpha,n+m-2} \right] \\ &+ P_{\mu_1\mu_2}\left[ \displaystyle\frac{\overline{X}-\overline{Y}}{S\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}} <-t^\ast_{1-\alpha,n+m-2} \right] \\ &= P_{\mu_1\mu_2}\left[ \displaystyle\frac{\displaystyle\frac{\overline{X}-\overline{Y}-(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}+\displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}}{S/\sqrt{(n+m-2)}} > t^\ast_{1-\alpha,n+m-2}\right] \\ &+ P_{\mu_1\mu_2}\left[ \displaystyle\frac{\displaystyle\frac{\overline{X}-\overline{Y}-(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}+\displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}}{S/\sqrt{(n+m-2)}} < -t^\ast_{1-\alpha,n+m-2}\right] \\ &= P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right] \\ &+ P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} < -t^\ast_{1-\alpha,n+m-2} \right] \end{aligned}

where Δ=(μ1−μ2)σ1n+1m\Delta = \displaystyle\frac{(\mu_1-\mu_2)}{\sigma\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}} and U is the SSE of the samples which is divided by the appropriate degrees of freedom to give a χ2\chi^2 distribution.

Note

Note that the power of a test can be obtained using the power.t.test function in R.

Sample Sizes for Two-sample One and Two-sided tt-tests​

The sample size should always satisfy the desired power.

Details​

Suppose we want to reject the H0H_0 with a pre-specified probability β1\beta_1 when μ1\mu_1 and μ2\mu_2 are true values of μ\mu. +For this, we need to select the sample size n and m so that β(μ1μ2)≥β1\beta_{(\mu_1\mu_2)} \geq \beta_1 i.e. find n and m which satisfies

Pμ1μ2[Z+ΔU/(n+m−2)>t1−α,n+m−2∗]P_{\mu_1\mu_2} \left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right]

for a two sample, one-sided t-test.

Similarly for a two sample, two-sided t-test we need to find n and m that satisfies

Pμ1μ2[Z+ΔU/(n+m−2)>t1−α,n+m−2∗]+Pμ1μ2[Z+ΔU/(n+m−2)<−t1−α,n+m−2∗]P_{\mu_1\mu_2}\left[ \displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} > t^\ast_{1-\alpha,n+m-2} \right] + P_{\mu_1\mu_2} \left[\displaystyle\frac{Z + \Delta}{\sqrt{U/(n+m-2)}} < -t^\ast_{1-\alpha,n+m-2} \right]

A Case Study In Power​

We want to compute power in analysis of covariance:

yij=μi+βxij+ϵij, i=1,2, j=1,…J,y_{ij}=\mu_i+\beta x_{ij}+\epsilon_{ij}, \ i=1, 2,\ j=1,\ldots J,

where ϵij∼N(0,σ2)\epsilon_{ij}\sim N(0,\sigma^ 2) are independent and identically distributed. +This can be done by simulation and can easily be expanded to other cases.

Handout​

If you want to compute a power analysis in analysis of covariance

yij=μi+βxij+ϵij, i=1,2, j=1,…J,y_{ij}=\mu_i+\beta x_{ij}+\epsilon_{ij}, \ i=1, 2,\ j=1,\ldots J,

where ϵij∼N(0,σ2)\epsilon_{ij}\sim N(0,\sigma^ 2) are independent identically distributed, then use simulation

To do this one needs to first define the task in more detail, along with what exactly is known and what the assumptions are.

Note

Note that there are only two groups, with intercepts μ1\mu_1 and μ2\mu_2. +The "power" will refer to the power of a test for μ1=μ2\mu_1=\mu_2 , i.e. we want to test whether the group means are equal, correcting for the effect of the continuous variable x.

In principle, the x-values will be either fixed a priori or they may be a random part of the experiment. +Here we will assume that the x-values are randomly selected in the range 20-30 (could e.g. be the ages of patients).

Since this is in the planning stage of the experiment, we also have a choice of the sample size within each group. +For convenience, the sample sizes are taken to be the same in each group, J so the total number of measurements will be n=2J. +We also need to decide at which levels of μ1\mu_1 and μ2\mu_2 the power is to be computed (but it is really only a function of the difference, μ1−μ2\mu_1-\mu_2 ).

The following pieces of R code can be saved into a file, ancovapow.r and then use the command:

source("ancovapow.r")

can be used to run the whole thing

The beginning of the command sequence merely consists of comments and definitions of parameter values. +These need to be changed for each case separately.

#
# ancovapow.r - power computations for analysis of covarariance
#             - one factor, two levels mu0, mu1
#             - one covariate x, x0 stores possible values from which a random set is chosen
#
# first set values of parameters
#
alpha <- 0.05
sigma <- 7.5        # the common standard deviation
x0 <- 20:30         # the set of x values
delta <- 10         # the difference in the means
mu0 <- 0            # the first mean
mu1 <- mu0+delta    # the second mean
slope <- 2.5        # the slope in the ancova
J <- 10             # the common sample size per factor level
n <- 2*J            # the total sample size
Nsim <- 40000       # the number of simulations for power computations

Rather than head straight for the ANCOVA, start with a simpler case, namely ignoring the covariate (x) and merely doing a regular two-sample, two-tailed t-test. +This should be reasonably similar to the ANCOVA power computations anyway.

#
# Next do the power computations just for a regular two-sided, two-sample t-test
# and use simulation
#
Y1 <- matrix(rnorm(J*Nsim,mu0,sigma),ncol=J)  # Simulate Nsim samples of size J, ea N(mu1,sigma^2)
Y2 <- matrix(rnorm(J*Nsim,mu1,sigma),ncol=J)  # Simulate Nsim samples of size J, ea N(mu2,sigma^2)
y1mn <- apply(Y1,1,mean)                      # compute all the simulated y1-means
y2mn <- apply(Y2,1,mean)                      # compute all the simulated y2-means
sy1 <- apply(Y1,1,sd)                         # compute all the simulated y1-std.devs
sy2 <- apply(Y2,1,sd)                         # compute all the simulated y2-std.devs
s <- sqrt(((J-1)*sy1^2+(J-1)*sy2^2)/(n-2))    # compute all the pooled std.devs
t <- (y1mn-y2mn)/(s*sqrt(1/J+1/J))            # compute all the Nsim t-statistics
i <- ifelse(abs(t)>qt(1-alpha/2,n-2),1,0)     # for ea t, compute 1=reject, 0=do not reject
powsim2 <- sum(i)/Nsim                        # the simulated power
cat("The simulated power is ",powsim2,"\n")

The above gave the simulated power. +In R there is a function to do the same computations and it is worth while to verify the code (and approach) by checking whether these give the same thing:

#
# Then compute the exact power for the t-test
#
pow2 <- power.t.test(delta=delta,sd=sigma,sig.level=alpha,n=J ,type=c("two.sample"),alternative=c("two.sided"))
cat("The exact power:\n")
print(pow2)

Finally, start setting up the code to do the ANCOVA simulations. +Note that for this we need to generate the xx -values. +In this example it is assumed that the xx -values are not under the control of the experimenter but arrive randomly, in the range from 20 to 30 (could e.g. be the age of participants in an experiment).

#
# Finally compute the power in the ANCOVA - note we already have simulated Y1, Y2-values but have not added the x-part yet
#
x1 <- matrix(sample(x0,Nsim*J,replace=T),ncol=J) # simulate x-values for y1
x2 <- matrix(sample(x0,Nsim*J,replace=T),ncol=J) # simulate x-values for y2
Y1 <- Y1+slope*x1
Y2 <- Y2+slope*x2
fulldat <- cbind(Y1,Y2,x1,x2) # a row now contains all y1, then all y2, then all x1, then all x2; Nsim rows

Rather than try to write code to do an ANCOVA, it is natural to use the R function lm to do this. +The "trick" below is to extract the pp -value from the summary command. +By defining a wrapper\verb|wrapper| function which takes a single line as an argument, it will subsequently be possible to use the apply\verb|apply| function to extract the pp -values using a one-line R command.

ancova.pval<-function(onerow){ # extract the ancova p-value for diff in means
  J <- length(onerow)/4
  n <- 2*J
  y <- onerow[1:n]                         # get the y-data from the row
  x <- onerow[(n+1):(2*n)]                 # get the x-data from the row
  grps <- factor(c(rep(1,J),rep(2,J)))     # define the groups
  sm <- summary(lm(y~x+grps))              # fit the ancova model
  pval <- sm$coefficients[3,4] # extract exactly the right thing from the summary command-the p-value for H0:mu1=mu2
  return(pval)
}

Everything has now been defined so it is possible to compute all the pp -values in a single command line:

pvec <- apply(fulldat,1,ancova.pval)
i2 <- ifelse(pvec<alpha,1,0)               # for ea test, compute 1=reject, 0=do not reject
ancovapow <- sum(i2)/Nsim                  # the simulated power
cat("The simulated ancova power is ",ancovapow,"\n")

When run, this script returns:

The simulated power is  0.803025
The exact power:

    Two-sample t test power calculation

             n = 10
         delta = 10
            sd = 7.5
     sig.level = 0.05
         power = 0.8049123
   alternative = two.sided

NOTE: n is number in each group

The simulated ANCOVA power is 0.775175.

It is seen that when the x-values are not included in any way (in particular, β=0\beta=0 ), the power is 180.5%. +However, this is not the correct model in the present situation. +Using the above value of β\beta. +Taking this into account, the power is actually a bit lower or 77.5%.

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Some Distributions Related to Normal

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Some Distributions Related to the Normal

The Normal and Sums of Normals​

The sum of independent normally distributed random variables is also normally distributed.

Details​

The sum of independent normally distributed random variables is also normally distributed. +More specifically, if X1∼N(μ1,σ12)X_1 \sim N(\mu_1, \sigma_{1}^2) and X2∼N(μ2,σ22)X_2 \sim N(\mu_2, \sigma_{2}^2) are independent, then X1+X2∼N(μ,σ2)X_1 + X_2 \sim N(\mu, \sigma^2), since μ=E[X1+X2]=μ1+μ2\mu = E \left[ X_1 + X_2 \right] = \mu_1 + \mu_2 and\ σ2=Var[X1+X2]\sigma^2 = Var \left[ X_1 + X_2 \right] with σ2=σ12+σ22\sigma^2 = \sigma_{1}^2 + \sigma_{2}^2 \ if X1X_1 and X2X_2 are independent.

Similarly

∑i=1nXi\displaystyle\sum_{i=1}^{n} X_i

is normal if X1,…,XnX_1, \ldots, X_n are normal and independent.

Examples​

Example: Simulating and plotting a single normal distribution.

Y∼N(0,1)Y \sim N(0,1)

library(MASS) # for truehist
par(mfcol=c(2,2))
y <- rnorm(1000) # generating 1000 N(0,1)
mn <- mean(y)
vr <- var(y)
truehist(y,ymax=0.5) # plot the histogram
xvec <-seq(-4,4,0.01) # generate the x-axis
yvec <- dnorm(xvec) # theoretical N(0,1) density
lines(xvec,yvec,lwd=2,col="red")
ttl <- paste("Simulation and theory N(0,1)\n", "mean=",round(mn,2), "and variance=",round(vr,2))
title(ttl)
Example: Sum of two normal distributions

Y1∼N(2,22)Y_1 \sim N(2, 2^2)

and

Y2∼N(3,32)Y_2 \sim N(3, 3^2)

y1 <- rnorm(10000,2,2) # N(2,2^2)
y2 <- rnorm(10000,3,3) # N(3, 3^2)
y <- y1+y2
truehist(y)
xvec <- seq(-10,20,0.01)
mn<-mean(y)
vr <- var(y)
cat("The mean is",mn,"\n")
cat("The variance is ",vr,"\n")
cat("The standard deviation is", sd(y), "\n")
yvec <- dnorm(xvec,mean=5,sd=sqrt(13)) # N() density
lines(xvec,yvec,lwd=2,col="red")
ttl <- paste("The sum of N(2,2^2) and N(3,3^2)\n", "mean=",round(mn,2), "and variance=", round(vr,2))
title(ttl)
Example Sum of nine normal distributions, all with μ=42\mu = 42 and σ2=22\sigma^2=2^2.
ymat <- matrix(rnorm(10000*9,42,2),ncol=9)
y <- apply(ymat,1,mean)
truehist(y)
mn <- mean(y)
vr <- var(y)
cat("The mean is",mn,"\n")
cat("The variance is ",vr,"\n")
cat("The standard deviation is",sd(y),"\n")
# plot the theoretical curve
xvec <- seq(39,45,0.01)
yvec <- dnorm(xvec,mean=5,sd=sqrt(13)) # N() density
lines(xvec,yvec,lwd=2,col="red")
ttl <- paste("The sum of nine N(42^2) \n", "mean=",round(mn,2), "and variance=",round(vr,2))
title(ttl)

The Chi-square Distribution​

If X∼N(0,1)X \sim N(0,1),then Y=X2Y = X^2 has a distribution which is called the chi-square distribution ( χ2\chi^2 ) on one degree of freedom. +This can be written as:

Y∼χ2Y \sim \chi^2

Fig. 34

Details​

Definition

If X1,X2,…,XnX_1, X_2, \ldots, X_n are i.i.d. N(0,1)N(0,1) then the distribution of Y=X12+X12+…+Xn2Y = X_1^2 + X_1^2 + \ldots + X_n^2 has a square ( χ2\chi^2 )distribution.

Sum of Chi-square Distributions​

Let Y1Y_1 and Y2Y_2 be independent variables. +If Y1=χν12Y_1 = \chi^2_{\nu_1} and Y2=χν22Y_2 = \chi^2_{\nu_2}, then the sum of these two variables also follows a chi-squared ( χ2\chi^2) distribution:

Y1+Y2=χν1+ν22Y_1 + Y_2 = \chi^2_{\nu_1+ \nu_2}

Fig. 35

Details​

Note

Recall that if

X1,…,Xn∼N(μ,σ2)X_1, \ldots, X_n \sim N (\mu, \sigma^2)

are i.i.d., then

∑i=1n(Xˉ−μσ)2=∑i=1n(Xˉ−μ)2σ∼χ2\displaystyle\sum_{i=1}^n \left ( \displaystyle\frac {\bar{X} - \mu} {\sigma}\right ) ^2= \displaystyle\sum_{i=1}^n \displaystyle\frac {\left ( \bar{X} - \mu\right ) ^2} {\sigma}\sim \chi^2

Sum of Squared Deviation​

If X1,⋯ ,Xn∼N(μ,σ2)X_1,\cdots,X_n \sim N(\mu,\sigma^2) i.i.d, then

∑i=1n(Xi−μσ)2∼χn2,\displaystyle\sum_{i=1}^n \left ( \displaystyle\frac{X_i-\mu}{\sigma} \right )^2 \sim \chi_{n}^2,

but we are often interested in

1n−1∑i=1n(Xi−Xˉ)2∼χn−12\displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^n (X_i-\bar{X})^2\sim \chi_{n-1}^2

Details​

Consider a random sample of Gaussian random variables, i.e. X1,⋯ ,Xn∼N(μ,σ2)X_1,\cdots,X_n \sim N(\mu,\sigma^2) i.i.d. +Such a collection of random variables have properties which can be used in a number of ways.

∑i=1n(Xi−μσ)2∼χn2\displaystyle\sum_{i=1}^n \left ( \displaystyle\frac{X_i-\mu}{\sigma} \right )^2 \sim \chi_{n}^2

but we are often interested in

1n−1∑i=1n(Xi−Xˉ)2∼χn−12\displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^n (X_i-\bar{X})^2\sim \chi_{n-1}^2

Note

A degree of freedom is lost because of subtracting the estimator of the mean as opposed to the true mean.

The correct notation is:

μ\mu := population mean

Xˉ\bar{X} := sample mean (a random variable)

xˉ\bar{x} := sample mean (a number)

The TT distribution​

If U∼N(0,1)U\sim N(0,1) and W∼χν2W\sim\chi^{2}_{\nu} are independent, then the random variable

T=UwνT=\displaystyle\frac{U}{\sqrt{\displaystyle\frac{w}{\nu}}}

has a distribution which we call the TT distribution on ν\nu degrees of freedom denoted T∼tνT \sim t_{\nu}.

Details​

Definition

If U∼N(0,1)U\sim N(0,1) and W∼χν2W\sim\chi^{2}_{\nu} are independent, then the random variable

T:=UwνT:=\displaystyle\frac{U}{\sqrt{\displaystyle\frac{w}{\nu}}}

has a distribution which we call the TT distribution on ν\nu degrees of freedom, denoted T∼tνT \sim t_\nu.

It turns out that if X1,…,Xn∼N(μ,σ2)X_1, \ldots,X_n \sim N(\mu,\sigma ^2) and we set:

Xˉ=1n∑i=1nXi\bar{X}=\displaystyle\frac{1}{n}\displaystyle\sum_{i=1}^n X_i

and

S=11−n∑i=1n(Xi−X)2S= \sqrt{\displaystyle\frac{1}{1-n}\displaystyle\sum_{i=1}^n (X_i-X)^2}

then

Xˉ−μS/n∼tn−1\displaystyle\frac{\bar{X}-\mu}{S/\sqrt{n}} \sim t_{n-1}

This follows from Xˉ\bar{X} and Σi=1N(Xi−Xˉ)2\Sigma_{i=1}^N(X_i-\bar{X})^2 being independent and Xˉ−μσ/n∼N(0,1)\displaystyle\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1), ∑(Xi−Xˉ)2σ2∼χn−12\displaystyle\sum \displaystyle\frac{(X_i-\bar{X})^2}{\sigma^2}\sim \chi_{n-1}^2.

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Test of Hypothesis, P Values and Related Concepts

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Test of Hypothesis, P Values and Related Concepts

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Test of Hypothesis, P Values and Related Concepts

The Principle of the Hypothesis Test​

The principle is to formulate a hypothesis and an alternative hypothesis, H0H_0 and H1H_1 respectively, and then select a statistic with a given distribution when H0H_0 is true and select a rejection region which has a specified probability (α)(\alpha) when H0H_0 is true.

The rejection region is chosen to reflect H1H_1, i.e. to ensure a high probability of rejection when H1H_1 is true.

Examples​

Example

Flip a coin to test

H0:P=12H_0: P = \displaystyle\frac {1} {2} vs H1:P≠12H_1: P \neq \displaystyle\frac {1} {2} \ Reject, if no heads or all heads are obtained in 6 trials, where the error rate is

P[Reject H0 when true]=P[All heads or all tails]=P[All heads]+P[All tails]=126+126=2⋅164=132<0.05\begin{aligned} P[ \text{Reject } H_0 \text{ when true}] &= P [\text{All heads or all tails}] \\ &= P[\text{All heads}] + P[\text{All tails}] \\ &= \displaystyle\frac {1} {2^6} + \displaystyle\frac {1} {2^6} \\ &= 2 \cdot \displaystyle\frac {1} {64} \\ &= \displaystyle\frac {1} {32} \\ &< 0.05 \end{aligned}

A variation of this test is called the sign test, which is used to test hypothesis of the form,

H0H_0 : true median equals zero using a count of the number of positive values.

The One Sided zz -test for normal mean​

Consider testing

H0:μ=μ0H_0: \mu = \mu_0

vs

H1:μ>μ0H_1: \mu > \mu_0

Where data x1,…,xnx_1, \ldots, x_n are collected as independent observations of X1,…,Xn∼N(μ,σ2)X_1, \ldots,X_n \sim N(\mu, \sigma^2) and σ2\sigma^2 is known. +If H0H_0 is true, then

xˉ∼N(μ0,σ2n)\bar {x} \sim N(\mu_0, \displaystyle\frac{\sigma^2}{n})

So,

Z=xˉ−μ0σn∼N(0,1)Z = \displaystyle\frac{\bar {x} - \mu_0}{\displaystyle\frac{\sigma} {\sqrt{n}}} \sim N(0,1)

It follows that,

P[Z>z∗]=αP[Z>z^\ast] = \alpha

Where

z∗=z1−αz^\ast = z_{1-\alpha}

So if the data x1,…,xnx_1, \ldots,x_n are such that,

z=xˉ−μ0σn>z∗z = \displaystyle\frac{\bar {x} - \mu_0}{\displaystyle\frac{\sigma} {\sqrt{n}}} > z^\ast

Then H0H_0 is rejected.

Examples​

Example

Consider the following data set: 47,42,41,45,4647, 42, 41, 45, 46.

Suppose we want to test the following hypothesis

H0:μ=42H_0 : \mu = 42

vs

H1:μ>42H_1 : \mu > 42

where σ=2\sigma = 2 is given. +The mean of the given data set can be calculated as

xˉ=44.2\bar {x} = 44.2

we can calculate zz by using following equation

z=xˉ−μσn=44.2−4225=2.20.8944=2.459\begin{aligned} z &= \displaystyle\frac{\bar {x} - \mu}{\displaystyle\frac{\sigma} {\sqrt{n}}} \\ &= \displaystyle\frac{44.2 - 42}{\displaystyle\frac{2} {\sqrt{5}}} \\ &= \displaystyle\frac{2.2}{0.8944} \\ &= 2.459 \end{aligned}

Here α=0,05\alpha = 0,05 so we have that

z∗=1.645z^\ast = 1.645

We obtain that 2,459>1,6452,459 > 1,645, i.e. z>z∗z> z^\ast and so H0H_0 is rejected with α=0.05\alpha = 0.05

The Two-sided zz -test for a normal mean​

z:=x‾−μ0sn∼N(0,1)z: =\displaystyle\frac{\overline{x}-\mu_0}{s\sqrt{n}} \sim N(0,1)

Details​

Consider testing H0:μ=μ0H_0: \mu=\mu_0 versus H1:μ≠μ0H_1: \mu \ne \mu_0 based on observation from X1‾,…,X‾∼N(μ,σ2)\overline{X_1},\dots, \overline{X} \sim N(\mu, \sigma^2) independent and identically distributed where σ2\sigma^2 is known. +If H0H_0 is true, then

Z:=x‾−μ0σn∼N(0,1)Z: = \displaystyle\frac{\overline{x}-\mu_0}{\sigma \sqrt{n}} \sim N(0,1)

and

P[∣z∣>z∗]=αP[|z| > z^\ast] = \alpha

with

z∗=z1−αz^\ast = z_{1-\alpha}

We reject H0H_0 if ∣z∣>z∗|z| > z^\ast. +If ∣z∣>z∗|z| > z^\ast is not true, then we cannot reject the H0H_0 .

Examples​

Example

In R, you may generate values to calculate the zz value. +The command that is generally used is: quantile

To illustrate:

z<-rnorm(1000,0,1) quantile(z,c(0.025,0.975)) 2.5% 97.5% -1.995806 2.009849

So, the zz value for a two-sided normal mean is ∣−1.99∣\left |-1.99 \right |.

The One-sided T-test for a Single Normal Mean​

Recall that if X1,…,Xn∼N(μ,σ2)X_1,\dots,X_n \sim N(\mu,\sigma^2) independent and identically distributed then

X‾−μS/n∼tn−1\displaystyle\frac{\overline{X}-\mu}{S/\sqrt{n}}\sim t_{n-1}

Details​

Recall that if X1,…,Xn∼N(μ,σ2)X_1,\ldots,X_n \sim N(\mu,\sigma^2) independent and identically distributed then

X‾−μS/n∼tn−1\displaystyle\frac{\overline{X}-\mu}{S/\sqrt{n}}\sim t_{n-1}

To test the hypothesis H0:μ=μ0H_0:\mu=\mu_{0} vs H1:μ>μ0H_1:\mu > \mu_{0} first note that if H0H_0 is true, then

T=X‾−μ0S/n∼tn−1T= \displaystyle\frac{\overline{X}-\mu_{0}}{S/\sqrt{n}} \sim t_{n-1}

so

P[T>t∗]=αP[T>t^{\ast}]=\alpha

if

t∗=tn−1,1−αt^{\ast}=t_{n-1,1-\alpha}

Hence, we reject H0H_0 if the data x1,…,xnx_1,\dots,x_n results in a a value of t:=x‾−μ0S/nt:=\displaystyle\frac{\overline{x}-\mu_0}{S/\sqrt{n}} such that t>t∗t>t^{\ast}, otherwise H0H_0 cannot be rejected.

Examples​

Example

Suppose the following data set (12,19,17,23,15,27) comes independently from a normal distribution and we need to test H0:μ=μ0H_0:\mu=\mu_0 vs H1:μ>μ0H_1:\mu>\mu_0. +Here we have n=6,x‾=18.83,s=5.46,μ0=18n=6,\overline{x}=18.83, s=5.46, \mu_0=18

so we obtain

t=x‾−μ0s/n=0.37t=\displaystyle\frac{\overline{x}-\mu_0}{s/\sqrt{n}}= 0.37

so H0H_0 cannot be rejected

In R, t∗t^{\ast} is found using qt(n-1,0.95) but the entire hypothesis can be tested using

t.test(x,alternative="greater",mu=<18)

Comparing Means from Normal Populations​

Suppose data are gathered independently from two normal populations resulting in x1,…,xnx_1,\dots,x_n and y1,…ymy_1,\dots y_m

Details​

We know that if

X1,…,Xn∼N(μ1,σ)X_1, \dots, X_n \sim N(\mu_1,\sigma)

Y1,…,Ym∼N(μ2,σ)Y_1, \dots, Y_m \sim N(\mu_2,\sigma)

are all independent then

Xˉ−Yˉ∼N(μ1−μ2,σ2n+σ2m)\bar{X}-\bar{Y} \sim N(\mu_1-\mu_2,\displaystyle\frac{\sigma^2}{n}+\displaystyle\frac{\sigma^2}{m})

Further,

∑i=1n(Xi−Xˉ)2σ2∼Xn−12\displaystyle\sum_{i=1}^{n} \displaystyle\frac{(X_i-\bar{X})^2}{\sigma^2} \sim X_{n-1}^{2}

and

∑j=1m(Yj−Yˉ)2σ2∼Xm−12\displaystyle\sum_{j=1}^{m} \displaystyle\frac{(Y_j-\bar{Y})^2}{\sigma^2} \sim X_{m-1}^{2}

so

Σi=1n(Xi−Xˉ)2+Σj=1m(Yj−Yˉ)2σ2∼Xn+m−22\displaystyle\frac {\Sigma_{i=1}^{n}(X_i-\bar{X})^2 + \Sigma_{j=1}^{m}(Y_j-\bar{Y})^2}{\sigma^2} \sim X_{n+m-2}^2

and it follows that

Xˉ−Yˉ−(μ1−μ2)S(1n+1m)∼tn+m−2\displaystyle\frac {\bar{X}-\bar{Y}-(\mu_1-\mu_2)}{S\sqrt{(\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m})}} \sim t_{n+m-2}

where

S=Σi=1n(X1−Xˉ)2+Σj=1m(Yj−Yˉ)2n+m−2S=\sqrt{\displaystyle\frac{\Sigma_{i=1}^{n}(X_1-\bar{X})^2+\Sigma_{j=1}^{m}(Y_j-\bar{Y})^2}{n+m-2}}

consider testing H0:μ1=μ2H_0:\mu_1=\mu_2 vs H1=μ1>μ2H_1=\mu_1>\mu_2. +Hence, if H0H_0

is true then the observed value

t=xˉ−yˉS1n+1mt=\displaystyle\frac{\bar{x}-\bar{y}}{S\sqrt{\displaystyle\frac{1}{n}+\displaystyle\frac{1}{m}}}

comes from a tt -test with n+m−2n+m-2 df and we reject H0H_0 if ∣t∣>t∗\left|t\right|>t^\ast. +Here,

S=∑i(xi−xˉ)2+∑j(yj−yˉ)2n+m−2S=\sqrt{\displaystyle\frac{\displaystyle\sum_{i}(x_i-\bar{x})^2+\displaystyle\sum_{j}(y_j-\bar{y})^2}{n+m-2}}

and t∗=tn+m−2,1−αt^\ast=t_{n+m-2,1-\alpha}

Comparing Means from Large Samples​

If X1,…XnX_1,\dots X_n and Y1,…YmY_1,\dots Y_m, are all independent (with finite variance) with expected values of μ1\mu_1 and μ2\mu_2 respectively, and variances of σ12\sigma_1^2,and σ22\sigma_2^2 respectively, then

X‾−Y‾−(μ1−μ2)σ12n+σ22m∼N(0,1)\displaystyle\frac{\overline{X}-\overline{Y}-(\mu_1-\mu_2)}{\sqrt{\displaystyle\frac{\sigma_1^2}{n}+\displaystyle\frac{\sigma_2^2}{m}}} \sim N(0,1)

if the sample sizes are large enough

This is the central limit theorem.

Details​

Another theorem (Slutzky) stakes that replacing σ12\sigma_1^2 and σ22\sigma_2^2 with S12S_1^2 and S22S_2^2 will result in the same (limiting) distribution

It follows that for large samples we can test

H0:μ1=μ2vsH1:μ1>μ2H_0: \mu_1=\mu_2 \qquad \text{vs} \qquad H_1:\mu_1 > \mu_2

by computing

z=x‾−y‾s12n+s22mz=\displaystyle\frac{\overline{x}-\overline{y}}{\sqrt{\displaystyle\frac{s_1^2}{n}+\displaystyle\frac{s_2^2}{m}}}

and reject H0H_0 if z>z1−αz>z_{1-\alpha}.

The P-value​

The pp -value of a test is an evaluation of the probability of obtaining results which are as extreme as those observed in the context of the hypothesis.

Examples​

Example

Consider a dataset and the following hypotheses

H0:μ=42H_0:\mu=42

vs.

H1:μ>42H_1:\mu>42

and suppose we obtain

z=2.3z=2.3

We reject H0H_0 since

2.3>1.645+z0.952.3>1.645+z_{0.95}

The pp -value is

P[Z>2.3]=1−Φ(2.3)P[Z>2.3]= 1-\Phi(2.3)

obtained in R using

1-pnorm(2.3) [1] 0.01072411

If this had been a two tailed test, then

P=P[∣Z∣>2.3]=P[Z<−2.3]+P[Z>2.3]=2⋅P[Z>2.3]\begin{aligned} P &= P[|Z|>2.3] \\ &= P[Z<-2.3]+P[Z>2.3] \\ &= 2\cdot P[Z>2.3] \end{aligned}

The Concept of Significance​

Details​

Two sample means are statistically significantly different if the null hypothesis H0:μ1=μ2H_0:\mu_1 = \mu_2, can be rejected. +In this case, one can make the following statements:

  • The population means are different.
  • The sample means are significantly different.
  • μ1≠μ2\mu_1 \ne \mu_2
  • xˉ\bar{x} is significantly different from yˉ\bar{y}.

But one does not say:

  • The sample means are different.
  • The population means are different with probability 0.950.95.

Similarly, if the hypothesis H0:μ1=μ2H_0: \mu_1 = \mu_2 cannot be rejected, we can say:

  • There is no significant difference between the sample means.
  • We cannot reject the equality of population means.
  • We cannot rule out.

But we cannot say:

  • The sample means are equal.
  • The population means are equal.
  • The population means are equal with probability 0.950.95.
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Multivariate to Power

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Multivariate to Power

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Data Vectors

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Data Vectors

The Plane​

Pairs of numbers can be depicted as points on a plane.

The plane is normally denoted by R2\mathbb{R}^2.

Details​

Pairs of numbers can be depicted as points on a plane.

Definition

A plane is a perfectly flat surface with no thickness and no end, it can extend forever in all directions. +It has two-dimensions, length and width. +We need two values to find a point on the plane.

Normally we talk about "the plane" (or "the xyxy-plane") as the collection of all pairs of numbers and denote it by

R2={(x,y):x,y∈R},\mathbb{R}^2 = \{ (x,y) : x,y \in \mathbb{R} \},

giving coordinates to each point. +The plane is also sometimes called The Cartesian coordinate system, named after its inventor, the French polymath René Descartes.

Examples​

Example

Plotting the point (2,4)(2,4) in the xyxy-plane using R:

plot(2,4,xlim=c(0,6),ylim=c(0,6),xlab="x",ylab="y",cex=2)
text(2,4,"(2,4)",pos=4,cex=2)

Additional points can be added using the points() function:

points(3,5, cex = 0.5) ## a point at (3,5)

If you have two sets of coordinates on a plane you, can calculate the distance between the two points and graph the line connecting the points.

Example

What is the distance between the two points (3,9)(3,9) and (5,1)(5,1)? +What is the distance between the 2 points (3,9) and (5,1)?

We will use the Pythagorean theorem:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}

We insert our values into the formula:

d=(5−3)2+(1−9)2d=\sqrt{(5-3)^{2}+(1-9)^{2}}

When we combine inside the parentheses we get:

d=(2)2+(−8)2d=\sqrt{(2)^{2}+(-8)^{2}}

Squaring both terms:

d=4+64d=\sqrt{4+64}

Then we take the square root:

d=68d=\sqrt{68}

The result:

d≈8.2462d \approx 8.2462

Simple Plots in R​

Graphing functions in R

  • plot() - plots a scatter plot (as a line plot)

  • points() - adds points to a plot

  • text() - adds text to a plot

  • lines() - adds lines to a plot

Fig. 3

Figure: Points on a plane, drawn with R.

Examples​

Example
plot(2,3)

gives a single plot and

plot(2,3, xlim=c(0,5), ylim=c(0,5))

gives a single plot but forces both axes to range from 0 to 5.

Example

The following R commands can be used to generate a plot with two points:

plot(1,2,xlim=c(0,5),ylim=c(0,5),xlab="x",ylab="y")
points(3,1)
text(1,2,"(1,2)",pos=4, cex=2)
text(3,1,"(3,1)",pos=4, cex=2)
Example

In this example, we plot three points. +The first two arguments of the plot function. +The third plot was added with the points are by including vectors with a length of 22 as the xx and yy arguments of the plot function. +The third point was added with the points function. +The second and third points were labeled using the text function and a line was drawn between them using the lines function.

plot(c(2,3),c(3,4),xlim=c(2,6),ylim=c(1,5),xlab="x",ylab="y")
points(4,2)
text(3,4,"(3,4)",pos=4, cex=2)
text(4,2,"(4,2)",pos=4, cex=2)
lines(c(3,4), c(4,2))

Note: Note that if you are unsure of what format the arguments of an R function needs to be, you can call a help file by typing ? before the function name (e.g. ?lines).

Data​

Data are usually a sequence of numbers, typically called a vector.

Details​

When we collect data these are one or more sequences of numbers, collected into data vectors. +We commonly think of these data vectors as columns in a table.

Examples​

Example

In R, if the command

x <- c(4,5,3,7)

is given, then x contains a vector of numbers.

Example

Create a function in R, give it a name Myfunction which takes the sum of x and y:

Myfunction <- function(x,y) { sum(x,y) }

If you input the vectors 1:3 and 4:7 into the function it will calculate the sum of x <- (1+2+3) and y <- (4+5+6+7) as follows:

> Myfunction(1:3,4:7)
[1] 28

Indices for a Data Vector​

If data are in a vector x, then we use indices to refer to individual elements.

Details​

If i is an integer then xix_i denotes the ithi^{th} element of xx.

Note: Although we do not distinguish (much) between row- and column vectors, usually a vector is thought of as a column. +If we need to specify the type of vector, row or column, then for vector xx, the column vector would be referred to as x′x' and the row vector as xTx^T.

(the transpose of the original).

Examples​

Example

If x=(4,5,3,7)x=(4,5,3,7) then x1=4x_1=4 and x4=7x_4=7

Example

How to remove all indices below a certain value in R?

> x <- c(1,5,8,9,4,16,12,7,11)

> x
[1]  1  5  8  9  4 16 12  7 11

> y <- x[x>10]

> y
[1] 16 12 11
Example

Consider a function that takes to vectors

a∈Rn,b∈Nma \in \mathbb{R}^n, b \in \mathbb{N}^m

as arguments with:

n≥mn \ge m

and:

1≤b1,…,bm≤n.1 \le b_1,\dots,b_m \le n.

The function returns the sum:

∑i=1mabi\sum_{i = 1}^m {a_b}_i

Long version:

> fn <- function(a,b) {
+ result <- sum(a[b])
+ return(result)
+ }

Short version:

fN <- function(a,b) sum(a[b])

Summation​

We use the symbol Σ\Sigma to denote sums.

In R, the sum function adds numbers.

Examples​

Example

If x=(4,5,3,7)x=(4,5,3,7)

then

∑i=14xi=x1+x2+x3+x4=4+5+3+7=19\sum_{i=1}^{4} x_i = x_1+x_2+x_3+x_4 = 4+5+3+7 = 19

and

∑i=24xi=x2+x3+x4=5+3+7=15.\sum_{i=2}^{4} x_i = x_2+x_3+x_4 = 5+3+7 = 15.

In R one can give the corresponding commands:

> x <- c(4,5,3,7)

> x
[1] 4 5 3 7

> sum(x)
[1] 19

> sum(x[2:4])
[1] 15
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Discrete Random Variables and the Binomial Distribution

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Discrete Random Variables and the Binomial Distribution

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Discrete Random Variables and the Binomial Distribution

Simple Probabilities​

Details​

Of all the possible 3-digit strings, (3x)\displaystyle\binom{3}{x} of them have xx heads. +So the probability of landing xx heads is (3x)px(1−p)3−x\displaystyle\binom{3}{x}p^x(1-p)^{3-x}.

Examples​

Example

Consider a biased coin which has probability pp of landing heads up. +If we toss this coin three independent times the possible outcomes are:

sequenceprobabilityNumber of headsHHHp⋅p⋅p=p33HHTp2(1−p)2HTHp2(1−p)2HTTp(1−p)21THHp2(1−p)2THTp(1−p)21TTHp(1−p)21TTT(1−p)30\begin{array}{c c c} \hline \\ \text{sequence} & \text{probability} & \text{Number of heads} \\ \hline \\ \text{HHH} & p \cdot p \cdot p=p^3 & 3 \\ \text{HHT} & p^2(1-p) & 2 \\ \text{HTH} & p^2(1-p) & 2 \\ \text{HTT} & p(1-p)^2 & 1 \\ \text{THH} & p^2(1-p) & 2 \\ \text{THT} & p(1-p)^2 & 1 \\ \text{TTH} & p(1-p)^2 & 1 \\ \text{TTT} & (1-p)^3 & 0 \\ \hline \end{array}
Example

It is also possible to aggregate these values into a table and describe only the number of heads obtained:

HeadsProbability p(x)0(1−p)313p(1−p)223p2(1−p)3p3\begin{array}{c c} \hline \\ \text{Heads} & \text{Probability}\ p(x) \\ \hline 0 & (1-p)^3 \\ 1 & 3p(1-p)^2 \\ 2 & 3p^2(1-p) \\ 3 & p^3 \\ \hline \end{array}

If we are only interested in the number of heads, then this table describes a probability mass function pp, namely the probability p(x)p(x) of every possible outcome xx of the experiment.

Example

Given that a year is 365 days and each day has the same probability of being someone's birthday. +What's the probability of at least 2 people sharing a birthday in a group of 25 people? +Now, calculating each of the possible outcomes could become very tedious. +That is calculating the odds that 2 people share a birthday, 3 people, 4 people, etc. +So instead we try to find out the odds that no one in the group shares a birthday and subtract those odds from 1 (100%).

First, let's look at the odds of only two people having distinct birthdays.

365365⋅364365=0.9973\displaystyle\frac{365}{365} \cdot \displaystyle\frac{364}{365} = 0.9973

Person one can be born on any day and the odds of having a distinct birthday are therefore 1. +The next person can be born on everyday but the 1 the other person was born on, so 364 days

Now let's say we add the 3rd person and calculate his/her odds of having a distinct birthday.

365365⋅364365⋅363365=0.9918\displaystyle\frac{365}{365} \cdot \displaystyle\frac{364}{365} \cdot \displaystyle\frac{363}{365} = 0.9918

This can also be rewritten as

365⋅364⋅3633653\displaystyle\frac{365 \cdot 364 \cdot 363}{365^3}

And we can do this on and on for all the 25 people we are interested in. +But that may also become a bit tedious. +So we use factorials instead. +So instead of doing

365⋅364⋅363⋯⋅34136525\displaystyle\frac{365 \cdot 364 \cdot 363 \dots \cdot 341}{365^{25}}

we do

365!340!36525=0.4313\displaystyle\frac{\displaystyle\frac{365!}{340!}}{365^{25}}=0.4313

Essentially the division of factorials here removes all the values that are less than 341341, making the two expressions completely equal. +Rewriting the expression like this is only done to simplify the task of typing it up for the computer to calculate, since most calculators don't understand …\dots.

Now remember this is the probability that no one shares a birthday. +So when we subtract this from 1 we get:

1−0.4313=0.56871-0.4313=0.5687

or roughly 57% odds of at least 2 people in a group of 25 sharing the same birthday.

This example is commonly known as the birthday paradox, since it seems to many people to be counter-intuitive that the probability of at least two people out of 25 sharing a birthday is higher than 50% when there are 365 birthdays to choose from.

Random Variables​

A random variable is a concept used to denote the outcome of an experiment before it is conducted.

Examples​

Example

Let XX denote the number of heads in a coin tossing experiment. +We can then talk about the probabilities of certain events such as obtaining two heads, i.e. X=2X=2. +We write this as:

P[X=2]=(n2)p2(1−p)n−2P[X=2] = \displaystyle{n \choose 2}p^2(1-p)^{n-2}

In general:

P[X=x]=(nx)px(1−p)n−xP[X=x] = \displaystyle{n \choose x}p^x (1-p)^{n-x}

where x=0,1,…,nx = 0,1,\dots,n

Handout​

Definition

A random variable, XX, is a function defined on a sample space, with outcomes in the set of real numbers.

It is simpler to think of a random variable as a symbol used to denote the outcome of an experiment before it is conducted.

Note

It is essential to distinguish between upper-case and lower-case letters when writing these probabilities - it makes no sense to write P[x=x]P[x=x].

Note

Random variables are generally denoted by upper case letters such as XX, YY and so on.

Note

To see how a random variable is a function, it is useful to consider the actual outcomes of two coin tosses. +These outcomes can be denoted {HH,HT,TH,TT}\{HH, HT, TH, TT\}. +Now consider a random variable XX which describes the number of heads obtained. +This random variable attributed 2 to the outcome HHHH and 0 to TTTT, i.e. XX is a function with X(HH)=2X(HH)=2, X(HT)=X(TH)=1X(HT)=X(TH)=1 and X(TT)=0X(TT)=0.

Simple Surveys With Replacement​

If we randomly draw individuals (with replacement), and ask a question with two possible answers (positive or negative), then the number of positive answers will come from a binomial distribution.

Examples​

Example

Suppose we are participating in a lottery. +We pick a number from a lottery bowl (a simple random sample). +We can put the number aside, or we can put it back into the bowl. +If we put the number back in the bowl, it may be selected more than once; +if we put it aside, it can be selected only one time.

Definition: When an element can be selected more than one time, we are sampling with replacement.

Definition: When an element can be selected only one time, we are sampling without replacement.

The Binomial Distribution​

If we toss a biased coin nn independent times, each with probability pp of landing heads up, then the probability of obtaining xx heads is

(nx)px(1−p)n−x\displaystyle{n \choose x}p^x (1-p)^{n-x}

Examples​

Example

Suppose we toss a coin, with probability pp of landing on heads nn times ,obtaining a sequence of H's (when it lands heads) and T's (when it lands tails). +Any sequence

HTH…HTHHHHTH \dots HTHHH

which has xx heads (HH) and n−xn-x tails (TT), has the probability px(1−p)n−xp^x(1-p)^{n-x}. +There are exactly (nx)\displaystyle\binom{n}{x} such sequences, so the total probability of landing xx heads in nn tosses is

(nx)px(1−p)n−x\displaystyle\binom{n}{x}p^x(1-p)^{n-x}

Example

Let the probability that a certain football club wins a match be equal to 0.4. +If the total number of matches played in the season is 30, what is the probability that the football club wins the match 10% of the time? +We first calculate the number of times a match was played and won by multiplying the percentage of wins by the number of matches played:

10% of 30 times = 3 times

We can now proceed to calculate the probability that they will win the match given that their probability of a winning is 0.4, if they play 3 times in a season. +This can be computed as follows:

(303)⋅(0.4)3⋅(1−0.4)30−3\displaystyle\binom{30}{3} \cdot (0.4)^3 \cdot (1-0.4)^{30-3}

=0.000265=0.000265

This can be calculated in R using the code below:

> dbinom(3,30,0.4)
[1] 0.0002659437

This is equal to the manual calculation using the binomial theorem.

Example

Suppose a youngster puts his shirt on by himself every day for five days. +The probability that he puts it on the right way each time is p=0.2p=0.2. +We let XX be a random variable that describes the number of times the youngster puts his shirt on the right way. +The youngster can either put the shirt on the wrong or the right way so XX follows the binomial distribution with the parameters p=0.2p=0.2 (the probability of a successful trial) and n=5n=5 (number of trials). +We can now calculate for example the probability that the youngster will put it on the right way for at least four days

Putting the shirt on the right way for at least four days, means that the youngster will either put it on the right way for either four or five days (at least four or more days of five days total). +We thus have to calculate the probability that the youngster will put his shirt on the right way for four and five days separately and then we add it together. +We can write this process as follows:

P(X≥4)=P(X=4)+P(X=5)=(54)⋅0.24⋅(1−0.2)5−4+(55)⋅0.25⋅(1−0.2)5−5=5⋅0.24⋅0.81+1⋅0.25⋅0.80=5⋅0.24⋅0.8+0.25⋅1=5⋅0.8⋅0.24+0.25=4⋅0.24+0.25=4⋅0.0016+0.00032=0.00672\begin{aligned} P(X\geq4) &= P(X=4) + P(X=5) \\ &= \displaystyle\binom{5}{4}\cdot0.2^4\cdot(1-0.2)^{5-4} + \displaystyle\binom{5}{5}\cdot0.2^5\cdot(1-0.2)^{5-5} \\ &= 5\cdot0.2^4\cdot0.8^1 + 1\cdot0.2^5\cdot0.8^0 \\ &= 5\cdot0.2^4\cdot0.8 + 0.2^5\cdot1 \\ & =5\cdot0.8 \cdot0.2^4 + 0.2^5 \\ &= 4\cdot0.2^4 + 0.2^5 \\ &= 4\cdot0.0016 + 0.00032 \\ &= 0.00672 \end{aligned}

The probability that the youngster will put his shirt on the right way for at least four out of five is thus approximately 0.7%.

This is possible to calculate in R in a several ways, either using the command dbinom or dbinom.

The command dbinom calculates

P(X=k)P(X = k)

and the command pbinom calculates

P(X≤k)P(X \leq k)

where kk is the number of successful trials.

If n is the number of trials and p is the probability of a successful trials then the commands are used by writing dbinom(k,n,p) and pbinom(k,n,p).

To calculate the probability that the youngster will put his shirt on the right way for at least four days of five we thus write the command:

> dbinom(4,5,0.2) + dbinom(5,5,0.2)
[1] 0.00672

which gives 0.00672.

This is the same as writing:

> dbinom(c(4,5),5,0.2)
[1] 0.00640 0.00032

or

> dbinom(4:5,5,0.2)
[1] 0.00640 0.00032

which give two separate numbers, 0.00640 and 0.00032 which can be added together to get 0.00672.

There is also a command to add them together for us:

> sum(dbinom(c(4,5),5,0.2))
[1] 0.00672

or

> sum(dbinom(4:5,5,0.2))
[1] 0.00672

They give the answer 0.00672.

The fourth way of calculating this in R is to use pbinom. +As said before pbinom calculates

P(X≤k)P(X \leq k)

where kk is the number of successful trials. +Here we want to calculate the probability that the youngster will put his shirt on the right way in 4 or 5 times (of 5 total) so the number of successful trials is 4 or greater. +That means we want to calculate

P(X≥4)P(X \geq 4)

which equals

1−P(X≤3)1 - P(X \leq 3)

We thus put kk as 3 and the R command will be:

> 1 - pbinom(3,5,0.2)
[1] 0.00672

which also gives 0.00672.

Example

In a certain degree program, the chance of passing an examination is 20%. +What is the chance of passing at most two exams if the student takes five exams?

Solution: In this problem, we compute the chance of a student passing, 0, 1 or 2 exams. +This is given by, In this problem, we compute the chance of a student passing, 0, 1 or 2 exams. +This is given by,

P(X=0 or 1 or 2)=(50)⋅0.20⋅0.85+(51)⋅0.21⋅0.84+(52)⋅0.22⋅0.83=1⋅0.20⋅0.85+5⋅0.21⋅0.84+10⋅0.22⋅0.83=0.32768+0.4096+0.2048=0.94208\begin{aligned} P(X=0 \text{ or }1 \text{ or }2) &= \displaystyle{5\choose 0} \cdot 0.2^0 \cdot 0.8^5 + \displaystyle{5\choose 1} \cdot 0.2^1 \cdot 0.8^4 + \displaystyle{5\choose 2} \cdot 0.2^2 \cdot 0.8^3 \\ &= 1 \cdot 0.2^0 \cdot 0.8^5 +5 \cdot 0.2^1 \cdot 0.8^4 +10 \cdot 0.2^2 \cdot 0.8^3 \\ &= 0.32768+0.4096+0.2048 \\ &= 0.94208 \end{aligned}

In the R console, we can use the command sum(dbinom(c(0:2),5,0.2)), which also gives 0.94208.

The same answer is obtained with

> dbinom(0,5,0.2)+dbinom(1,5,0.2)+dbinom(2,5,0.2)
[1] 0.94208

and with

> pbinom(2,5,0.2)
[1] 0.94208
Example

Consider the probability of someone jumping off a cliff is 0.35. +Suppose we randomly selected four individuals to participate in the cliff jumping activity. +What is the chance that exactly one of them will jump off the cliff? +Consider a scenario where person A jumps but B, C and D refuse.

=P(A=jump,B=refuse,C=refuse,D=refuse)=P(A=jump)⋅P(B=refuse)⋅P(C=refuse)⋅P(D=refuse)=(0.35)⋅(0.65)⋅(0.65)⋅(0.65)=0.35⋅(0.65)3=0.096\begin{aligned} &= P (A = \text{jump}, B = \text{refuse}, C = \text{refuse}, D = \text{refuse}) \\ &= P (A =\text{jump}) \cdot P (B = \text{refuse}) \cdot P (C = \text{refuse}) \cdot P (D = \text{refuse}) \\ &= (0.35) \cdot (0.65) \cdot (0.65) \cdot (0.65) \\ &= 0.35 \cdot (0.65)^3 = 0.096 \end{aligned}

But there are three other scenarios (where persons B, C, or D are the only person to jump). +In each of these cases, the probability is again 0.096. +These four scenarios exhaust all the possible ways that exactly one of the four people jumps:

=4⋅0.35⋅(0.65)3=0.38\begin{aligned} &= 4 \cdot 0.35 \cdot (0.65)^3 \\ &= 0.38 \end{aligned}

In the R console we can use the command dbinom(1,4,0.35) which gives the answer as 0.384475.

General Discrete Probability Distributions​

A general discrete probability distribution can be described by a list of all possible outcomes and associated probabilities.

Details​

A general discrete probability distribution is described by the possible outcomes

x1,x2,…x_1, x_2, \ldots

and associated probabilities, denoted by p1,p2,…p_1, p_2, \ldots or p(x1),p(x2),…p(x_1), p(x_2),\ldots. +If a random variable XX has this distribution, then we can write:

P[X=xi]=p(xi)=piP[X=x_i] = p(x_i)= p_i

or, in general:

P[X=x]=p(x)P[X=x] = p(x)

where it is understood that p(x)=0p(x) = 0 if xx is not one of these xix_i.

Examples​

Example

If XX is the number of heads ( H\text{H} ) before obtaining the first tail ( T\text{T} ) when tossing an unbiased coin 4 independent times, then the possible basic outcomes are:

TossIn binary1 2 3 4H before T0000H H H T30010H H T H20011H H T T20100H T H H10101H T H T10110H T T H10111H T T T11000T H H H01001T H H T01010T H T H01011T H T T01100T T H H01101T T H T01110T T T H01111T T T T0\begin{array}{c c c} \hline & \text{Toss} & \\ \text{In binary} & 1\ 2\ 3\ 4 & \text{H before T} \\ \hline \\ 0000 & \text{H H H T} & 3 \\ 0010 & \text{H H T H} & 2 \\ 0011 & \text{H H T T} & 2 \\ 0100 & \text{H T H H} & 1 \\ 0101 & \text{H T H T} & 1 \\ 0110 & \text{H T T H} & 1 \\ 0111 & \text{H T T T} & 1 \\ 1000 & \text{T H H H} & 0 \\ 1001 & \text{T H H T} & 0 \\ 1010 & \text{T H T H} & 0 \\ 1011 & \text{T H T T} & 0 \\ 1100 & \text{T T H H} & 0 \\ 1101 & \text{T T H T} & 0 \\ 1110 & \text{T T T H} & 0 \\ 1111 & \text{T T T T} & 0 \\ \hline \end{array}

Since the coin is unbiased, each of these has the same probability of occurring. +We can now count sequences to find the number of possibilities of a particular number of heads, H\text{H}, before a tail in 4 coin tosses and thus obtain the corresponding probabilities as:

Number of tosses before a headsProbabilityp(x)0816=121416=142216=1831164116\begin{array}{c r} \hline \\ \text{Number of tosses before a heads} & \text{Probability} \\ & p(x) \\ \hline \\ 0 & \displaystyle\frac{8}{16} = \displaystyle\frac{1}{2} \\ 1 & \displaystyle\frac{4}{16} = \displaystyle\frac{1}{4} \\ 2 & \displaystyle\frac{2}{16} = \displaystyle\frac{1}{8} \\ 3 & \displaystyle\frac{1}{16} \\ 4 & \displaystyle\frac{1}{16} \\ \hline \end{array}

The Expected Value or Population Mean​

The expected value is the sum of the possible outcomes, weighted with the respective probabilities (discrete variable). +Think of this in terms of an urn full of marbles, each labelled with number.

Details​

If the possible outcomes are x1,x2,…x_1, x_2, \dots with probabilities p1,p2,…p_1, p_2, \dots then the expected value is:

μ=x1⋅p1+x2⋅p2+…\mu=x_1 \cdot p_1+x_2 \cdot p_2 + \ldots

The fact that this is the only sensible definition of an expected value follows from considering random draws from a finite population where there are nin_i possibilities of obtaining the value xix_i. +If we set n=∑xin=\sum x_i and pi=ni/np_i=n_i/n then the expected value above is the simple average of all the numbers in the original population.

In the case of the binomial distribution with nn trials and success probability pp it turns out that:

μ=n⋅p\mu=n \cdot p

If XX is the corresponding random variable, we denote this quantity by E[X]E[X].

Examples​

Example

If we toss a fair coin ten independent times, we expect on average np=10⋅12=5np=10 \cdot \displaystyle\frac{1}{2}= 5 heads.

Example

Toss a fair die and pay 6060 if a six comes up and nothing otherwise. +The expected outcome is:

56⋅0+16⋅60=10\displaystyle\frac{5}{6} \cdot 0 + \displaystyle\frac{1}{6} \cdot 60 = 10

Example

In Las Vegas, a particular sports bet has about a 30% chance of winning. +If the bet wins, the bettor will win 15 dollars. +If the bet loses, the bettor will lose 10 dollars. +The expected return of placing one of these bets is -2.50 dollars.

Detailed calculation:

15⋅0.3−10⋅0.7=−2.515 \cdot 0.3 - 10 \cdot 0.7 = - 2.5

Example

Class starts at 8:00 and the last bus that will get you to class on time leaves at 7:30. +The teacher has a policy that if you are late to class 6 of the 30 classes, then she drops your final grade by 1/10 points. +You know that if you set your alarm for 7:15, you miss the 7:30 bus approximately every fourth time, but if you set it for 7:10, you'll only miss the bus approximately every eighth time. +If you set it for 7:00, you'll only miss the bus every one hundredth time.

Part A: Assuming you try to go to class every time, can you expect to have your grade dropped in the following scenarios?

  1. You set your alarm for 7:15 throughout the duration of the class.
  2. You set your alarm for 7:15 until you reach 5 missed classes, then switch to 7:10.
  3. You set your alarm for 7:15 until you reach 5 missed classes, then switch to 7:00.

Part B: What is your expected grade in the course, assuming you would have had a 7/10 without the late penalty, and:

  1. You would never choose the first alarm-clock strategy and you would most likely choose scenario 2 (let's say 9/10 times), but there's a small chance you might choose the 3rd strategy (let's say 1/10 times).
  2. You would never choose the first alarm-clock strategy and you would most likely choose scenario 3 (let's say 9/10 times), but there's a small chance you might choose the 2nd strategy (let's say 1/10 times)

Answers:

A1: Let XX be a random variable which denotes to be the number of times we make it to class on-time. +With the alarm set to 7:15 we expect to make it to class on-time: A1 - Let's call X our random variable, which we want to be the number of times we make it to class on-time. +With the alarm set to 7:15 we expect to make it to class on-time:

E[X]=30⋅(1−14)=452=22.5E[X]=30\cdot (1-\displaystyle\frac{1}{4}) = \displaystyle\frac{45}{2}=22.5

Your grade would most likely be dropped.

A2: First we need to see how many classes we go to before we reach the 5-late-classes threshold:

E[X]=n⋅(1−14)=n−5E[X] = n \cdot (1 - \displaystyle\frac{1}{4}) = n - 5

E[X]=n((1−14)−1)=−5E[X] = n ((1 - \displaystyle\frac{1}{4}) - 1) = - 5

E[X]=n=−5−14E[X] = n = \displaystyle\frac{- 5}{- \displaystyle\frac{1}{4}}

E[X]=n=201=20E[X] = n = \displaystyle\frac{20}{1} = 20

So, the night before our 21st class, you get worried and change alarm-clock strategies. +If you set it at 7:15 for the rest of the course (10 classes), you will be on time:

E[X]=15+10⋅(1−18)=954=23.75E[X] = 15 + 10 \cdot (1 - \displaystyle\frac{1}{8}) = \displaystyle\frac {95}{4} = 23.75

Your grade would most likely be dropped.

A3: If you instead start setting the alarm clock for 7:00 for the rest of the course, you will be on time:

E[X]=15+10⋅(1−1100)=2179≈24.11E[X] = 15 + 10 \cdot (1 - \displaystyle\frac{1}{100}) = \displaystyle\frac{217}{9} \approx 24.11

You're grade would most likely NOT be dropped

Part B: This seems to contain errors In Part A, we calculated the mean of several binomial distributions that described the expected number of days that you will arrive on-time to class. +Each distribution corresponded to a different alarm-setting scenario. +In this part, we are describing a different binomial distribution. +It describes your expected grade. +Therefore, the grade is the outcome nn, weighted by the probability of you choosing the particular alarm-clock setting procedure:

1−E[X]=0⋅6+0.9⋅6+0.1⋅7=6.11 - E[X] = 0 \cdot 6 + 0.9 \cdot 6 + 0.1 \cdot 7 = 6.1

1−E[X]=0⋅6+0.1⋅6+0.9⋅7=6.91 - E[X] = 0 \cdot 6 + 0.1 \cdot 6 + 0.9 \cdot 7 = 6.9

Note that the probabilities of these three choices (0 + 0.9 + 0.1) must equal 1, since these are the only three choices defined.

The Population Variance​

The (population) variance, for a discrete distribution, is:

σ2=E[(X−μ)2]=(x1−μ)2p1+(x2−μ)2p2+…\sigma^2 = E\left[ \left ( X-\mu \right ) ^2 \right ] = (x_1 - \mu)^2 p_1 + (x_2 - \mu)^2 p_2 + \dots

where it is understood that the random variable XX has this distribution and μ\mu is the expected value.

In the case of the binomial distribution, it turns out that:

σ2=np(1−p)\sigma^2 = np(1 - p)

Details​

Definition

If μ\mu is the expected value, then the variance of a discrete distribution is defined as

σ2=(x1−μ)2p1+(x2−μ)2p2+…\sigma ^2=(x_1 - \mu)^2 p_1 + (x_2 - \mu)^2 p_2 + \ldots

If a random variable XX has associated probabilities, pi=P[X=xi]p_i=P[X=x_i], then one can equivalently write:

σ2=Var[X]=E[(X−μ)2]\sigma^2 = Var[X]=E\left [ \left ( X - \mu \right ) ^ 2\right ]

Examples​

Example

In the case of the binomial distribution, it turns out that:

σ2=np(1−p)\sigma^2 = np(1 - p)

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Functions

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Functions

Functions of a Single Variable​

A function describes the relationship between variables.

Examples:

f(x)=x2f(x) = x^2

y=2+3⋅x4y = 2+3\cdot x^4

Fig. 4

Details​

Functions are commonly used in statistical applications, to describe relationships.

Definition

A function describes the relationship between variables. +A variable yy is described as a function of a variable xx by completely specifying how yy can be computed for any given value of xx.

An example could be the relationship between a dose level and the response to the dose.

The relationship is commonly expressed by writing either f(x)=x2f(x) = x^{2} or y=x2y = x^2.

Usually names are given to functions, i.e. to the relationship itself. +For example, ff might be the function and f(x)f(x) could be its value for a given number xx. +Typically f(x)f(x) is a number but ff is the function, but the sloppy phrase "the function f(x)=2x+4f(x)=2x+4" is also common.

Examples​

Example

f(x)=x2f(x) = x^2 or y=x2y = x^2 specifies that the computed value of yy should always be x2x^2, for any given value of xx.

Functions in R​

A function can be defined in R using the function command:

Fig. 5

Ranges and Plots in R​

Functions in R can commonly accept a range of values and will return a corresponding vector with the outcome.

Examples​

Example
f <- function(x) {return(x*12)}
x <- seq (-5,5,0,1)
y <- f(x)
plot {(x,y) type= 'l'}

Plotting Functions​

In statistics, the function of interest is commonly called the response function. +If we write y=f(x)y=f(x), the outcome yy is usually called the response variable and xx is the explanatory variable. +Function values are plotted on vertical axis while xx values are plotted on horizontal axis. +This plots yy against xx.

Examples​

Example

The following R commands can be used to generate a plot for function y=2+3xy= 2+3x:

x <- seq(0:10)
g <- function(x) {
+ yhat <- 2+3*x
+ return(yhat)
+ }

x <- seq(0,10,0.1)
y <- g(x)
plot(x,y,type="l", xlab="x",ylab="y")

Functions of Several Variables​

Examples​

Example

z=2x+3y+4v=t2+3xw=t2+3b⋅x\begin{aligned} z &= 2x+3y+4\\ v &= t^2+3x\\ w &= t^2+3b \cdot x\end{aligned}

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Indices and the apply Commands in R

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Indices and the Apply Commands in R

Giving Names to Elements​

We can name elements of vectors and data frames in R using the names command.

Examples​

Example
> X <- c(41, 3, 73)
> names(X) <- c("One", "Two", "Three")

View the results by simply typing X and the output of X is given as follows:

> X
  One   Two Three
   41     3    73

With this we can refer to the elements by name as well as locations using:

> X[1]
One
 41

> X["Three"]
Three
   73

Regular Matrix Indices and Naming​

A matrix is a table of numbers. +Typical matrix indexing: mat[i,j], mat[1:2,] etc. +A matrix can have row and column names Indexing with row and column names: mat["a","B"].

Details​

Definition

A matrix is a (two-dimensional) table of numbers, indexed by row and column numbers.

Note

Note that a matrix can also have row and column names, so that the matrix can be indexed by its names rather than numbers.

Examples​

Example

Consider a matrix with 2 rows and 3 columns. +Consider extracting first element (1,2)(1,2), then all of line 2 and then columns 2-3 in an R session:

> mat <- matrix(1:6,ncol=3)

> mat
     [,1] [,2] [,3]
[1,]    1    3    5
[2,]    2    4    6

> mat[1,2]
[1] 3

> mat[2,]
[1] 2 4 6

> mat[,2:3]
     [,1] [,2]
[1,]    3    5
[2,]    4    6

Next, consider the same matrix, but give names to the rows and columns. +The rows will get the names aa and bb and the columns will be named AA, BB and CC.

The entire R session could look like this:

> mat <- matrix(1:6,ncol=3)

> dimnames(mat) <- list(c("a","b"),c("A","B","C"))

> mat
  A B C
a 1 3 5
b 2 4 6

> mat["b",c("B","C")]
B C
4 6

The apply Command​

apply(mat,2,sum)\verb|apply(mat,2,sum)| -- applies the sum function within each column

apply(mat,1,mean)\verb|apply(mat,1,mean)| -- computes the mean within each row

The tapply Command​

Commonly one has a data vector and another vector of the same length giving categories for the measurements. +In this case one often wants to compute the mean or variance (or median etc) within each category. +To do this we use the tapply command in R.

Examples​

Example
> z <- c(5,7,2,9,3,4,8)
> i <- c("m","f","m","m","f","m","f")

A. Find the sum within each group

> tapply(z,i,sum)
 f  m
18 20

B. Find the sample sizes

> tapply(z,i,length)
f m
3 4

C. Store outputs and use names

> n <- tapply(z,i,length)

> n
f m
3 4

> n["m"]
m
4

Logical Indexing​

A logical vector consists of TRUE\verb|TRUE| (1) or FALSE\verb|FALSE| (0) values. +These can be used to index vectors or matrices.

Examples​

Example
> i <- c("m","f","m","m","f","m","f")

> z <- c(5,7,2,9,3,4,8)

> i=="m"
[1]  TRUE FALSE  TRUE  TRUE FALSE  TRUE FALSE

> z[i=="m"]
[1] 5 2 9 4

> z[c(T,F,T,T,F,T,F)]
[1] 5 2 9 4

Lists, Indexing Lists​

A list is a collection of objects. +Thus, data frames are lists.

Examples​

Example
> x <- list(y=2,z=c(2,3),w=c("a","b","c"))

> x[["z"]]
[1] 2 3

> names(x)
[1] "y" "z" "w"

> x["w"]
$w
[1] "a" "b" "c"

> x$w
[1] "a" "b" "c"
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More on Algebra

Some Squares​

If aa and bb are real numbers, then

(a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2

Details​

If aa and bb are real numbers, then:

(a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2

This can be proven formally with the following argument:

(a+b)2=(a+b)(a+b)=(a+b)a+(a+b)b=a2+ba+ba+b2=a2+2ab+b2\begin{aligned} (a+b)^2 &=& (a+b)(a+b)\\ &=&(a+b)a+(a+b)b\\ &=& a^2+ba+ba+b^2\\ &=& a^2+2ab+b^2\end{aligned}

Pascal's Triangle​

Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle:

111121⋮⋮⋮⋮⋮\begin{array}{ccccc} & & 1 & &\\ & 1 & & 1&\\ 1 & & 2 & & 1\\ \vdots \quad \vdots && \vdots && \vdots \quad \vdots \end{array}

Details​

n=0:1n=1:11n=2:121n=3:1331\begin{array}{ccccccccc} n=0: & & & & &1& & & \\ n=1: & & & &1& &1& & \\ n=2: & & &1& &2& &1& \\ n=3: & &1& &3& &3& &1 \end{array}

To build Pascal's triangle, start with 1 at the top, and then continue placing numbers below it in a triangular pattern. +Each number is just the two numbers above it added together (except for the edges, which are all 1).

Examples​

Example

The following function in R gives you the Pascal's triangle for n=0n=0 to n=10n=10.

> fN <- function(n) formatC(n, width=2)

> for (n in 0:10) {
+ cat(fN(n),":", fN(choose(n, k = -2:max(3, n+2))))
+ cat("\n")
+ }

 0 :  0  0  1  0  0  0
 1 :  0  0  1  1  0  0
 2 :  0  0  1  2  1  0  0
 3 :  0  0  1  3  3  1  0  0
 4 :  0  0  1  4  6  4  1  0  0
 5 :  0  0  1  5 10 10  5  1  0  0
 6 :  0  0  1  6 15 20 15  6  1  0  0
 7 :  0  0  1  7 21 35 35 21  7  1  0  0
 8 :  0  0  1  8 28 56 70 56 28  8  1  0  0
 9 :  0  0  1  9 36 84 126 126 84 36  9  1  0  0
10 :  0  0  1 10 45 120 210 252 210 120 45 10  1  0  0

Changing the numbers in the line for(n in 0:10) will give different portions of the triangle.

Factorials​

We define the factorial of an integer n as

n!=n⋅(n−1)⋅(n−2)⋅…⋅3⋅2⋅1n!= n\cdot(n-1) \cdot(n-2)\cdot \ldots \cdot 3 \cdot 2 \cdot 1

For convenience we define 0!0! to be 1.

Details​

Definition

We define the factorial of an integer n as:

n!=n⋅(n−1)⋅(n−2)⋯…⋅3⋅2⋅1n!= n\cdot(n-1) \cdot(n-2)\cdots \ldots \cdot 3 \cdot 2 \cdot 1

Examples​

Example

Suppose you have six apples: {a,b,c,d,e,f}\{a, b, c, d, e, f\} and you want to put each one into a different apple basket: {1,2,3,4,5,6}\{1,2,3,4,5,6\}.

For the first basket you can choose from 6 apples {a,b,c,d,e,f}\{a, b, c, d, e,f\}, and for the second basket you have then 5 apples to choose from and so it goes for the rest of the baskets, so for the last one you only have 1 apple to choose from.

The end result would then be 6!=6⋅5⋅4⋅3⋅2⋅1=7206! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720 possible allocations.

This could also be calculated in R with the factorial function:

> factorial(6)
[1] 720

Combinations​

The number of different ways one can choose a subset of size xx from a set of nn elements is determined using the following calculation:

(nx)=n!x!(n−x)!\displaystyle{n \choose x}= \displaystyle\frac{{n!}}{{x!\left( {n - x} \right)!}}

Details​

Definition

A combination is an un-ordered collection of distinct elements.

Suppose we want to toss a coin nn times. +In each toss we obtain head (H) or tail (T) resulting in a sequence of H, T, T, H, ..., T.

How many of these possible sequences contain exactly xx tails? +There are nn positions in the sequence, we can choose xx of these in (nx)\displaystyle\binom{n}{x} ways and put our Ts in those positions. +If the probability of landing tails is pp, then each one of these sequences with exactly xx tails has probability px(1−p)n−xp^x(1-p)^{n-x} +So the total probability of landing exactly xx tails in nn independent tosses is:

(nx)=n!x!(n−x)!\displaystyle{n \choose x}= \displaystyle\frac{{n!}}{{x!\left( {n - x} \right)!}}

Examples​

Example

Consider tossing a coin four times:

(a) How many times will this experiment result in exactly 2 tails? +There are a total of 16 possible sequences of heads and tails from 4 tosses. +These can simply all be written down to answer a question like this

We get two tails in 6 of these tosses. +We can explicitly write the corresponding combinations of two tails as follows:

HHTT HTHT HTTH THTH TTHH THHT

(b) How many times you will end up with 1 tail? The answer is 4 times and the output can be written as:

HHHT HTHH THHH HHTH

The case of a single tail is easy: The single tail can come up in any one of four positions.

The Binomial Theorem​

(a+b)n=∑x=0n(nx)axbn−x(a+b)^n = \sum_{x=0}^n \displaystyle{n \choose x} a^xb^{n-x}

Details​

If aa and bb are real numbers and nn is an integer then the expression (a+b)n(a+b)^n can be expanded as:

(a+b)n=an+(n1)an−1b+(n2)an−2b+…+(nn−1)abn−1+bn(a+b)^n = a^n+ \displaystyle{n \choose 1}a^{n-1}b + \displaystyle{n \choose 2}a^{n-2}b^ + \ldots + \displaystyle{n \choose n-1}ab^{n-1}+b^n

(a+b)n=σi=1n(nx)axbn−x(a+b)^n = \sigma_{i=1}^n \displaystyle{n \choose x}a^xb^{n-x}

This can be seen by looking at (a+b)n(a+b)^n as a product of nn parentheses and multiply these by picking one item (aa or bb) from each. +If we picked aa from xx parentheses and bb from (n−x)(n-x), then the product is axbn−xa^x b^{n-x}. +We can choose the xx aa's in a total of (nx)\displaystyle\binom{n}{x} ways so the coefficient of axbn−xa^x b^{n-x} is (nx)\displaystyle\binom{n}{x}.

Examples​

Example

Since

(a+b)n=∑x=0n(nx)axbn−x,(a+b)^n = \sum_{x=0}^n \displaystyle{n \choose x} a^xb^{n-x},

it follows that

2n=(1+1)n=∑x=0n(nx)2^n = (1+1)^n = \sum_{x=0}^n \displaystyle{n \choose x}

i.e.

2n=(n0)+(n1)+(n2)…+(nn)2^n = \displaystyle{n \choose 0} + \displaystyle{n \choose 1} + \displaystyle{n \choose 2}\ldots + \displaystyle{n \choose n}

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Numbers, Arithmetic and Basic Algebra

Natural Numbers​

The positive integers are called natural numbers. +These numbers can be added, multiplied together and so forth.

Notation: N={1,2,3,4,… }\mathbb{N}=\{1,2,3,4,\dots \}

Subtraction and division are not defined on these numbers.

An arbitrary element of N\mathbb{N} is most commonly denoted by i,j,ni, j, n, or mm, but any symbol can be used.

Details​

Definition

The set of positive integers is usually denoted by N\mathbb{N}, i.e. N={1,2,3,4,… }\mathbb{N}=\{1,2,3,4,\dots \} and is called the set of natural numbers. +In some cases the number zero is included as a natural number, but here we will use the symbol N0\mathbb{N}_0 to denote the integers 0, 1, 2 and up.

Within this set of numbers it is possible to add and multiply numbers together. +Arithmetic operations are denoted by ++ for addition and ⋅\cdot (or ×\times) for multiplication. +A natural number can also be raised to the power of a natural number, e.g. 35=3⋅3⋅3⋅3⋅33^5=3\cdot 3\cdot 3\cdot 3\cdot 3 or in general mn=m⋅m⋅…⋅mm^n=m\cdot m \cdot \ldots \cdot m ( nn times).

When stating general properties of the natural numbers one needs to use symbols to indicate that the property holds for an arbitrary number. +It is not enough to just write the property for a few numbers. +For example, to declare that one can interchange numbers in a sum, it is not enough to say 4+3=3+44+3=3+4, but one must explicitly state "the addition operator has the property that any two natural numbers, n, m∈Nn,\ m\in \mathbb{N} satisfy n+m=m+nn+m=m+n".

An arbitrary element of N\mathbb{N} is most commonly denoted by i, j, ni,\ j,\ n,or mm, but any symbol, a, b, c,…a,\ b,\ c, \ldots, can be used.

Several rules of arithmetic apply (some by definition, others can be derived) such as

ab=baa+b=b+aa+bc=a+(bc)a(b+c)=ab+ac(a+b)+c=a+(b+c)(ab)c=a(bc)\begin{aligned} ab &= ba \\ a+b &= b+a \\ a+bc &= a+(bc) \\ a(b+c) &= ab+ac \\ (a+b)+c &= a+(b+c) \\ (ab)c &= a(bc) \end{aligned}

Subtraction and division are not generally defined. +In addition, we define one integer, nn, to the power of another, mm, to mean nn multiplied by itself mm times: nm=n⋅n⋅…⋅n⏟mn^m=\underbrace{n\cdot n \cdot \ldots \cdot n}^{m}.

Definition

The power is an operator just like addition and multiplication, and is defined to have higher priority than the other two.

Examples​

Example

If we have x=4x=4 and y=2y= 2 and want to evaluate:

xy+yxx^y+y^x

then we replace the values of x and y in the expression, and evaluate it, taking care to observe the correct order of operations:

42+24=16+16=324^2+2^4=16+16=32

Starting With R​

Download R from the R website: http://www.r-project.org/

Look at on-line information on R, and take the tutor-web R tutorial: http://tutor-web.net/stats/stats240.1

Simple R commands:

  • Assignment:

    x <- 2

  • Arithmetic:

    2*5+4

Details​

To assign values to a variable in R one can use <- or =. +however, these are NOT equivalent. +Using the equals sign is confusing and therefore not recommended.

Examples​

Example

Assigning values to a variable:

x <- 2
y <- 3
z <- x+y
Example: Viewing assigned values

Type the name, i.e. z, to view the assigned value.

> z
[1] 5

The Integers​

The set of positive and negative integers:

Z:={…,−2,−1,0,1,2,… }\mathbb{Z} := \{\dots, -2, -1, 0, 1, 2, \dots \}

Details​

Definition

The set of all integers is denoted by Z\mathbb{Z}, i.e.

Z:={…,−2,−1,0,1,2,… }\mathbb{Z} := \{\dots, -2, -1, 0, 1, 2, \dots \}

Note

Note that within this set it is possible to subtract as well as add and multiply. +Within this set we cannot, however, in general, perform division.

When preforming multiple mathematical operations within the same equation, i.e. 79−8⋅379 - 8\cdot 3, there is a conventional order for which the operations must be performed.

Definition

The conventional order of operations for equations with multiple mathematical operations is referred to as an operator precedence.

Examples​

Example

To compute 79−8⋅379 - 8\cdot 3 start by multiplying and then subtracting:

79−8⋅3=79−24=5579 - 8\cdot 3 = 79-24 = 55

Example

To compute

15−(24+36)15 - (24 + 36)

we first note that the parentheses (brackets) imply a precedence; +anything inside brackets should be evaluated first. +Thus, we first add 3636 to 2424 and then we subtract that from 1515:

15−(24+36)=15−60=−4515 - (24+36) = 15 - 60 = - 45

Note that the answer is a negative number.

Example

Simple arithmetic in R is easily done at the command prompt:

> 79-8*3
[1] 55

> 15-(24+36)
[1] -45

Rational Numbers​

Rational numbers are fractions denoted pq\displaystyle\frac{p}{q}, where pp and qq are integers. +We can simplify fractions if the numerator and denominator contain common terms.

Details​

Fig. 1

Definition

Rational numbers are fractions denoted p/qp/q, where pp and qq are integers. +The set of all rational numbers is usually denoted Q\mathbb{Q}.

Note

Note that every integer is a rational number (obtained by taking q=1q=1).

We can simplify fractions if the numerator and denominator contain common terms. +When the rationals are ordered on to a line there are points missing, i.e. there are "gaps"; +for example there is no rational number p/qp/q such that (p/q)2=2(p/q)^2=2.

Examples​

Example

26=22⋅3=13\displaystyle\frac{2}{6} = \displaystyle\frac{2}{2 \cdot 3} = \displaystyle\frac{1}{3}

The rational numbers can be put in order along a line as in the figure.

Example

As an elaborate example of a fraction, consider the evaluation of the quantity:

23+2513+12\displaystyle\frac{\displaystyle\frac{2}{3}+\displaystyle\frac{2}{5}}{\displaystyle\frac{1}{3}+\displaystyle\frac{1}{2}}

Example

Evaluate:

23+2513+12\displaystyle\frac{\displaystyle\frac{2}{3}+\displaystyle\frac{2}{5}}{\displaystyle\frac{1}{3}+\displaystyle\frac{1}{2}}

Solution: We can either start by calculating the numerator:

23+25\displaystyle\frac{2}{3}+\displaystyle\frac{2}{5}

or the denominator:

13+12\displaystyle\frac{1}{3}+\displaystyle\frac{1}{2}

Here we choose to start with the numerator. +The first step is to make the two fractions in the numerator have a common denominator.

We can either find the least common denominator or use the product of the two denominators. +Here they are the same number, 15.

So the first step is:

23⋅55+25⋅33=displaystyle2⋅53⋅5+2⋅35⋅3=1015+615\displaystyle\frac{2}{3} \cdot \displaystyle\frac{5}{5} + \displaystyle\frac{2}{5} \cdot \displaystyle\frac{3}{3} = \\displaystyle\frac{2 \cdot 5}{3 \cdot 5} + \displaystyle\frac{2 \cdot 3}{5 \cdot 3} = \displaystyle\frac{10}{15} + \displaystyle\frac{6}{15}

Now it is possible to add the two fractions, which is the second step:

10+615=1615\displaystyle\frac{10+6}{15} = \displaystyle\frac{16}{15}

Next, the same process has to be performed for the original denominator.

With the same method (LCM - least common multiple) we get:

1⋅23⋅2+1⋅32⋅3=26+36=56\displaystyle\frac{1 \cdot 2}{3 \cdot 2} + \displaystyle\frac{1 \cdot 3}{2 \cdot 3} = \displaystyle\frac{2}{6} + \displaystyle\frac{3}{6} = \displaystyle\frac{5}{6}

Then the total answer is:

161556=1615⋅65=9675=96/375/3=3225\displaystyle\frac{\displaystyle\frac{16}{15}}{\displaystyle\frac{5}{6}} = \displaystyle\frac{16}{15} \cdot \displaystyle\frac{6}{5} = \displaystyle\frac{96}{75} = \displaystyle\frac{96/3}{75/3} = \displaystyle\frac{32}{25}

We can see that in the last step of the equation, the factor has been simplified. +To do this we use factoring. +Here we obtain:

9675=3⋅323⋅25\displaystyle\frac{96}{75} = \displaystyle\frac{3 \cdot 32}{3 \cdot 25}

We can now remove 33, or the multiplier, as it is on both sides of the fraction. +So we have:

3225=2525+725=3225\displaystyle\frac{32}{25} = \displaystyle\frac{25}{25} + \displaystyle\frac{7}{25} = \displaystyle\frac{32}{25}

In step 1 above we used Cross-Multiplication.

Definition: Cross-Multiplication is when we multiple the numerator by the reciprocal of the denominator.

So in this case we rewrite

161556\displaystyle\frac{\displaystyle\frac{16}{15}}{\displaystyle\frac{5}{6}}

or

1615÷56\displaystyle\frac{16}{15} \div \displaystyle\frac{5}{6}

as

1615⋅65\displaystyle\frac{16}{15} \cdot \displaystyle\frac{6}{5}

As you can see all we are doing is turning

56\displaystyle\frac{5}{6}

upside down; +and multiplying it with

1615\displaystyle\frac{16}{15}

This gives

9675\displaystyle\frac{96}{75}

In some cases it is possible to draw a square root of a fraction s=pqs = \displaystyle\frac{p}{q}, i.e. find a number r∈Qr\in \mathbb{Q} such that r2=sr^2=s. +The square root is denoted r\sqrt{r}.

Example

Consider the expression

(19⋅24)+(15⋅25)\left(\sqrt{\displaystyle\frac{1}{9}} \cdot 2^4\right) + (\displaystyle\frac{1}{5} \cdot \sqrt{25})

To evaluate this expression, first consider separately the two parts on each side of the plus symbol.

The first part is

(19⋅24)\left(\sqrt{\displaystyle\frac{1}{9}} \cdot 2^4\right)

and the second part is

(15⋅25)(\displaystyle\displaystyle\frac{1}{5} \cdot \sqrt{25})

In addition, by definition of root,

19=13\sqrt{\displaystyle\frac{1}{9}} = \displaystyle\frac{1}{3}

First part:

(19⋅24)=13⋅16=163\left(\sqrt{\displaystyle\frac{1}{9}} \cdot 2^4\right) = \displaystyle\frac{1}{3} \cdot 16 = \displaystyle\frac{16}{3}

Second part:

(15⋅25)=15⋅5=1(\displaystyle\frac{1}{5} \cdot \sqrt{25}) = \displaystyle\frac{1}{5} \cdot 5 = 1

Finally, add the first part and the second part:

163+1=193\displaystyle\frac{16}{3} + 1 = \displaystyle\frac{19}{3}

Example

Consider the following fraction example, to be solved step by step:

42+(14l⋅53)26÷15\displaystyle\frac{\displaystyle\frac{4}{2} + (\displaystyle\frac{1}{4l} \cdot \displaystyle\frac{5}{3})}{\displaystyle\frac{2}{6} \div \displaystyle\frac{1}{5}}

First we need to be aware of operator precedence, sometimes called BODMAS (brackets, multiplication/division, then addition/subtraction).

(14⋅53)=512(\displaystyle\frac{1}{4} \cdot \displaystyle\frac{5}{3}) = \displaystyle\frac{5}{12}

After solving the bracket we can proceed with adding

42\displaystyle\frac{4}{2}

to

512\displaystyle\frac{5}{12}

as there is no other action left for the nominator of the main fraction.

So:

42+512\displaystyle\frac{4}{2} + \displaystyle\frac{5}{12}

When adding fractions together we first have to find a common denominator, in this case 12 would work as

2⋅6=122\cdot6=12

So we multiply both the numerator and the denominator of that fraction by 6 and then add the two numerators of the fractions together, keeping the same denominator.

42+512=4⋅62⋅6+512=2412+512=2912\displaystyle\frac{4}{2} + \displaystyle\frac{5}{12} = \displaystyle\frac{4 \cdot 6}{2 \cdot 6} + \displaystyle\frac{5}{12} = \displaystyle\frac{24}{12} + \displaystyle\frac{5}{12} = \displaystyle\frac{29}{12}

Now we have the top half of the fraction solved. +We then proceed with dividing the two fractions of the bottom half. +When dividing fractions we use the so called cross multiplication technique. +This arithmetic trick is derived from the fact that if you divide a fraction by its duplicate you get 1. +If you multiple a fraction by its reciprocal (it's reverse) you also get 1. +Like so:

12÷12=1\displaystyle\frac{1}{2} \div \displaystyle\frac{1}{2}=1

and

12⋅21=1\displaystyle\frac{1}{2} \cdot \displaystyle\frac{2}{1}=1

These functions always provide the same result and therefore we can turn the fraction we are dividing by upside down and multiply it to the other fraction as that is usually much easier.

We can therefore rewrite

26÷15\displaystyle\frac{2}{6} \div \displaystyle\frac{1}{5}

as

26⋅51=106\displaystyle\frac{2}{6} \cdot \displaystyle\frac{5}{1} = \displaystyle\frac{10}{6}

We've now solved both halves of the original fraction and can therefore proceed to solve it, again with the cross multiplication technique as fractions are after all just divisions:

2912÷106=2912⋅610=174120\displaystyle\frac{29}{12} \div \displaystyle\frac{10}{6} = \displaystyle\frac{29}{12} \cdot \displaystyle\frac{6}{10} = \displaystyle\frac{174}{120}

Now

174120\displaystyle\frac{174}{120}

is a pretty bad looking fraction and we'd preferably like to simplify it. +To do this we use factoring.

Definition: Factoring essentially means to break a number done into its smallest factors or multipliable prime numbers.

In this case we get

2⋅3⋅292⋅3⋅20\displaystyle\frac{2 \cdot 3 \cdot 29}{2 \cdot 3 \cdot 20}

These are the smallest prime numbers that can multiply together into 174 and 120 respectively.

A way of doing this in your head is by first dividing both numbers (174 and 120) by 2. +Which gives us:

2⋅872⋅60\displaystyle\frac{2 \cdot 87}{2 \cdot 60}

and then dividing those numbers (87 and 60) by 3, since they can't be divided by 2. +Dividing by 3 gives you

3⋅293⋅20=2920\displaystyle\frac{3 \cdot 29}{3 \cdot 20} = \displaystyle\frac{29}{20}

which is a lot nicer than

174120\displaystyle\frac{174}{120}

The reasoning behind this factoring simplification is that we can remove multipliers if they are on both sides of a fraction. +This is because the result of a fraction where the numerator and the denominator are the same is always 1. +Like so:

11=1\displaystyle\frac{1}{1}=1

or

22=1\displaystyle\frac{2}{2}=1

or

33=1\displaystyle\frac{3}{3}=1

The final answer therefore is

42+(14⋅53)26÷15=2920\displaystyle\frac{\displaystyle\frac{4}{2} + (\displaystyle\frac{1}{4} \cdot \displaystyle\frac{5}{3})}{\displaystyle\frac{2}{6} \div \displaystyle\frac{1}{5}} = \displaystyle\frac{29}{20}

The Real Line​

Some obvious numbers are not fractions. +The set of numbers making up the real line is denoted by the symbol R\mathbb{R}.

Fig. 2

Figure: The diagonal of a rectangle with unit side lengths of 2\sqrt{2}. +Note that 2\sqrt{2} is not a fraction.

Details​

Some obvious numbers, which commonly occur, are not fractions. +These are in between the rational numbers (fractions) on the number line. +Filling in the missing points to obtain a continuum results in the set of real numbers.

Denoted by R\mathbb{R} the entire set of real numbers which corresponds to filling in the missing pieces of the number line, so to speak.

Examples​

Example

If CC is the circumference of a circle and DD is the diameter and we define π=CD\pi = \displaystyle\frac{C}{D} then π\pi is not a fraction.

Example

One example of a non-fraction is the number e (Euler's number) which can be defined by:

e=∑n=0∞1n!e = \sum_{n=0}^{\infty }\displaystyle\frac{1}{n!}

Example

If you have a right triangle with unit side length, what is the length of its hypotenuse and what class of numbers does it belong to?

An isosceles triangle is defined as having adjacent and opposite sides of same length, connected by a 90∘90^{\circ} angle. +Unit side length of these, refers to a side length of 1.

As we have a 90∘90^{\circ} angle, we can use Pythagoras' theorem:

a2+b2=c2a^2+b^2=c^2

With

a=adjacenta=\text{adjacent}

b=oppositeb=\text{opposite}

c=hypotenusec=\text{hypotenuse}

So with

a,b=1:a,b=1:

c2=12+12c^2=1^2+1^2

c2=1+1c^2=1+1

c2=2c^2=2

We take the square root to get cc:

c=2c=\sqrt{2}

Now that we answered the first part of the question, it needs to be defined, which class of number 2\sqrt{2} belongs to. +2\sqrt{2} is an irrational number, and belongs thereby to the set of real numbers R\mathbb{R}. +Real numbers can be imagined as points on an infinitely long line, which is also called the real line.

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Polynomials

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Polynomials

The General Polynomial​

The general polynomial:

p(x)=a0+a1x+a2x2+⋯+anxnp(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{n}x^{n}

The simplest:

p(x)=ap(x)=a

Details​

Definition

A polynomial describes a specific function consisting of linear combinations of positive integer powers of the explanatory variable.

The general form of a polynomial is:

p(x)=a0+a1x+a2x2+⋯+anxnp(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{n}x^{n}

The simplest of these is the constant polynomial

p(x)=ap(x)=a

The Quadratic​

The general form of the quadratic (parabola) is:

p(x)=ax2+bx+cp(x) = ax^2 + bx + c

The simplest quadratic is

p(x)=x2p(x) = x^2

Fig. 6

Figure: Parabolas: Quadratic functions

Details​

The quadratic polynomial of the form p(x)=ax2+bx+cp(x) = ax^2 + bx + c describes a parabola when points (x,y)(x,y) with y=p(x)y = p(x) are plotted. +The simplest parabola is p(x)=x2p(x) = x^2 (Fig. a) which is always non-negative p(x)≥0p(x)\geq 0 and p(x)=0p(x)=0 only when x=0x=0.

Note

Note that p(−x)=p(x)p(-x) = p(x) since (−x)2=x2(-x)^2= x^2.

If the leading coefficient is negative, then the parabola is concave (fig. b) but if it's positive the parabola is convex (fig. a). +This is sometimes used to describe a response function.

The Cubic​

The general form of a cubic polynomial is:

p(x)=ax3+bx2+cx+dp(x)=ax^3 + bx^2 + cx + d

Fig. 7

Figure:

p(x)=x3−20x2−30x−4p(x)=x^3-20x^2-30x-4

The Quartic​

The general form of the quartic polynomial is

p(x)=ax4+bx3+cx2+dx+ep(x) = ax^4 + bx^3 + cx^2 + dx + e

Fig. 8

Figure: The general shape.

Here we used the following equation

y=x4−x3−7x2+x+6y=x^4-x^3-7x^2+x+6

Solving the Linear Equation​

If the value of yy is given and we know that xx and yy are on a specific line so that y=a+bxy = a + bx, then we can find the value of xx.

Details​

If a value of yy is given, and we know that xx and yy lie on a specific straight line, so that y=a+bxy = a + bx, then we can find the value of xx by considering y=a+bxy = a+bx as an equation to be solved for xx, since yy, aa and bb are all known.

The general solution is found through the following steps:

  1. Equation: y=a+bxy = a + bx

  2. Subtract aa from both sides.

  3. y−a=bxy-a = bx

  4. bx=y−abx=y-a

  5. Divide by bb on both sides if bb is not equal to 0.

  6. x=1b(y−a)x=\displaystyle\frac{1}{b}(y-a)

Roots of the Quadratic Equation​

The general solution of ax2+bx+c=0ax^2 + bx + c = 0 is given by x=−b±b2−4ac2ax = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Details​

Suppose we want to solve ax2+bx+c=0ax^2 + bx + c = 0, where a≠0a \neq 0.

The general solution is given by the formula

x=−b±b2−4ac2a,x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac}}{2a},

if b2−4ac≥0b^2 - 4ac \geq 0.

On the other hand, if b2−4ac<0b^2-4ac<0, the quadratic equation has no real solution.

Examples​

Example

Solve x2−3x+2=0x^2 - 3x + 2 = 0

Putting this into the context of the formulation ax2+bx+c=0ax^2+bx+c=0, the constants are: a=1,b=−3,c=2a = 1, b = -3, c = 2.

Inserting this into the formula for the roots gives:

x=−(−3)±(−3)2−4⋅1⋅22⋅1x=3±9−82x=3±12x=3+12 or 3−12x=42 or 22x=2 or 1\begin{aligned} x &= \displaystyle\frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \\ x &= \displaystyle\frac{3 \pm \sqrt{9 - 8}}{2} \\ x &= \displaystyle\frac{3 \pm \sqrt{1}}{2} \\ x &= \displaystyle\frac{3 + 1}{2} \text{ or } \displaystyle\frac{3 - 1}{2} \\ x &= \displaystyle\frac{4}{2} \text{ or } \displaystyle\frac{2}{2} \\ x &= 2 \text{ or } 1 \end{aligned}
Example

Find the roots of the following polynomial

3x4+14x2+153x^{4} + 14x^{2} + 15

We can use the quadratic equation to solve for the roots of this polynomial if we substitute a variable for x2x^{2}. +Let's use the letter aa:

3a2+14a+153a^{2} + 14a + 15

We then plug the constants in to the quadratic equation.

x=−(14)±142−4⋅3⋅152⋅3x = \displaystyle\frac{-(14) \pm \sqrt{14^{2} - 4 \cdot 3 \cdot 15}}{2 \cdot 3}

which simplifies to:

−(14)±196−1806\displaystyle\frac{-(14) \pm \sqrt{196 - 180}}{6}

which equals −53-\displaystyle\frac{5}{3} (using the ++ sign) and −3-3 (using the −- sign).

Then, since we substituted a for x2x^2 we need to take the square root of these values to get the roots of the polynomial.

So,

x1,2=±−53x_{1,2} = \pm \sqrt{-\displaystyle\frac{5}{3}}

and

x3,4=±3x_{3,4} = \pm \sqrt{3}

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Simple Data Analysis in R

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Simple Data Analysis in R

Entering Data. Data Frames​

Several methods exist to enter data into R:

  1. Enter directly:

    x <- c(4,3,6,7,8)

  2. Read in a single vector

    x <- scan("filename")

  3. Use:

    x <- read.table("file address")

Details​

The most direct method will not work if there are a lot numbers; +therefore, the second method is to read in a single vector by

x <- scan("filename")

where filename = text string, either a full path name or refers to a file in the working directory.

The scan() command returns a vector, but the read.table() command returns a data frame, which is a rectangular table of data whose columns have names. +A column can be extracted from a data frame, e.g., with x <- dat$a where dat is the name of the data frame and a is the name of a column.

Note

Note that for read.table("file address"), file address refers to the location of the file. +Thus, it can be the URL or the complete file directory depending on where the table is stored.

Examples​

Example

Below are three examples using R code to enter data

  1. x <- c(4,3,6,7,8)

  2. x <- scan("lecture 70.txt")

  3. x <- read.table("http&#58;&#47;&#47;notendur&period;hi&period;is/~gunnar/kennsla/alsm/data/set115.dat", header=T)

Histograms​

A histogram is a graphical display of tabulated frequencies, shown as bars.

In R use the command hist(x).

Fig. 9

Examples​

A histogram is a graphical display of tabulated frequencies, shown as bars.

Bar Charts​

The bars in a bar chart usually correspond to frequencies in categories and are therefore kept apart.

Fig. 10

Details​

A bar chart is similar to the histogram but is used for categorical data.

Mean, Standard Error, Standard Deviations​

Details​

The most familiar measure of central tendency is the arithmetic mean.

Definition

An arithmetic mean is the sum of the values divided by the number values, typically expressed as:

yˉ=Σi=1nyin\bar{y} = \displaystyle\frac{\Sigma_{i=1}^{n} y_i}{n}

Definition

The sample variance is a measure of the spread of a set of values from the mean value:

s2=1n−1∑i=1n(xi−xˉ)2s^2 = \displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^{n}(x_i - \bar{x} )^2

The sample standard deviation is more commonly used as a measure of the spread of a set of values from the mean value.

Definition

The standard deviation is the square root of the variance and may be expressed as:

s=1n−1∑i=1n(xi−xˉ)2s = \sqrt{\displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^{n}(x_i - \bar{x} )^2}

Definition

The standard error is a method used to indicate the reliability of the sample mean:

SEyˉ=s2nSE_{\bar{y}} = \sqrt{\displaystyle\frac{s^2}{n}}

If a vector x in R contains an array of numbers then: mean(x) returns the average, bar{x};sd(x); returns the standard deviation, s;var(x); returns the variance, s^2. +We may also want to use several other related operations in R: median(x), the median value in vector x; range(x) which lists the range: max(x)-min(x).

If the variable x contains discrete categories, table(x) returns counts of the frequency in each category.

Scatter Plots and Correlations​

If we have paired explanatory and response data we are often interested in seeing if a relationship exists between them. +To do this, we first plot the data in a scatter plot.

Fig. 11

Figure: Scatter plot showing the length-weight relationship of fish species X.

Data source: Marine Resource Institution - Iceland.

Details​

A first step in analyzing data is to prepare different plots. +The type of variable will determine the type of plot. +For example, when using a scatter plot both the explanatory and response data should be continuous variables

The equation for the Pearson correlation coefficient is:

rx,y=Σi=1n(xi−xˉ)(yi−yˉ)Σi=1n(xi−xˉ)2Σi=1n(yi−yˉ)2,r_{x,y} = \displaystyle\frac{\Sigma_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\Sigma_{i=1}^{n}(x_i - \bar{x})^2 \Sigma_{i=1}^{n}(y_i - \bar{y})^2},

where xˉ\bar{x} and yˉ\bar{y} are the sample means of the x- and y-values.

The correlation is always between -1 and 1.

Examples​

The following R commands can be used to generate a scatter plot for vectors x and y

Example
plot(x,y)
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Independence, Expectations and the Moment Generating Function

Independent Random Variables​

Recall that two events, AA and BB, are independent if,

P[A∩B]=P[A]P[B]P [A \cap B] = P[A] P[B]

Since the conditional probability of AA given BB is defined by:

P[A∣B]=P[A∩B]P[B]P [A|B] = \displaystyle\frac {P [A \cap B]} {P[B]}

We see that AA and BB are independent if and only if

P[A∣B]=P[A](when P[B]>0)P[A|B] = P[A] \quad (\text{when } P [B] > 0 )

Two continuous random variables, XX and YY, are similarly independent if,

P[X∈A,Y∈B]=P[X∈A]P[Y∈B]P [X \in A, Y \in B] = P [X \in A] P[Y \in B]

Details​

Two continuous random variables, XX and YY, are similarly independent if,

P[X∈A,Y∈B]=P[X∈A]P[Y∈B]P [X \in A, Y \in B] = P [X \in A] P[Y \in B]

Now suppose XX has p.d.f. fXf_X, and YY has p.d.f. fYf_Y. +Then,

P[X∈A]=∫AfX(x)dx,P [X \in A] = \displaystyle\int_{A} f_X (x) dx,

P[Y∈B]=∫BfY(y)dyP [Y \in B] = \displaystyle\int_{B} f_Y (y) dy

So XX and YY are independent if:

P[X∈,Y∈B]=∫AfX(x)dx∫BfY(y)dy=∫AfX(x)(∫BfY(y)dy)dx=∫A∫BfX(x)fY(y)dydx\begin{aligned} P [X \in, Y \in B] &= \displaystyle\int_{A} f_X (x) dx \displaystyle\int_{B} f_Y (y) dy \\ &= \displaystyle\int_{A}f_X (x) (\displaystyle\int_{B} f_Y (y) dy) dx \\ &= \displaystyle\int_{A}\displaystyle\int_{B} f_X (x)f_Y (y) dydx \end{aligned}

But, if ff is the joint density of XX and YY then we know that

P[X∈A,Y∈B]P [X \in A, Y \in B]

∫A∫Bf(x,y)dydx\displaystyle\int_{A}\displaystyle\int_{B} f (x,y) dydx

Hence XX and YY are independent if and only if we can write the joint density in the form of,

f(x,y)=fX(x)fY(y)f(x,y) = f_X (x)f_Y (y)

Independence and Expected Values​

If XX and YY are independent random variables then E[XY]=E[X]E[Y]E[XY]=E[X]E[Y].

Further, if XX and YY are independent random variables then E[g(X)h(Y)]=E[g(X)]E[h(Y)]E[g(X)h(Y)]=E[g(X)]E[h(Y)] is true if gg and hh are functions in which expectations exist.

Details​

If XX and YY are random variables with a joint distribution function f(x,y)f(x,y), then it is true that for h:R2→Rh:\mathbb{R}^2\to\mathbb{R} we have

E[h(X,Y)]=∫∫h(x,y)f(x,y)dxdyE[h(X,Y)]=\displaystyle\int\displaystyle\int h(x,y)f(x,y)dxdy

for those hh such that the integral on the right exists

Suppose XX and YY are independent continuous random variables, then

f(x,y)=fX(x)fY(y)f(x,y) = f_X (x) f_Y (y)

Thus,

E[XY]=∫∫xyf(x,y)dxdy=∫∫xyfX(x)fY(y)dxdy=∫xfX(x)dx∫yfY(y)dy=E[X]E[Y]\begin{aligned} E[XY] &= \displaystyle\int\displaystyle\int xy f (x,y) dxdy \\ &= \displaystyle\int\displaystyle\int xy f_X (x) f_Y (y) dxdy \\ &= \displaystyle\int xf_X (x) dx \displaystyle\int yf_Y (y) dy \\ &= E[X] E[Y] \end{aligned}
Note

Note that if XX and YY are independent then E[h(X)g(Y)]=E[h(X)]E[g(Y)]E[h(X) g(Y)] = E [h(X)] E[g(Y)] is true whenever the functions hh and gg have expected values.

Examples​

Example

Suppose X,Y∈U(0,2)X,Y \in U (0,2) are independent identically distributed then,

fX(x)={12if 0≤x≤20otherwisef_X(x) = \begin{cases} \displaystyle\frac{1}{2} & \text{if } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases}

and similarly for fYf_Y.

Next, note that,

f(x,y)=fX(x)fY(y)={14if 0≤x,y≤20otherwisef(x,y) = f_X(x) f_Y(y) = \begin{cases} \displaystyle\frac{1}{4} & \text{if } 0 \leq x,y \leq 2 \\ 0 & \text{otherwise} \end{cases}

Also note that f(x,y)≥0f(x,y) \geq 0 for all (x,y)∈R2(x,y) \in \mathbb{R}^2 and

∫∫f(x,y)dxdy=∫02∫0214dxdy=14.4=1\displaystyle\int\displaystyle\int f(x,y)dxdy = \displaystyle\int_{0}^{2}\displaystyle\int_{0}^{2} \displaystyle\frac {1}{4} dxdy = \displaystyle\frac {1}{4}.4 = 1

It follows that

E[XY]=∫∫f(x,y)xydxdy=∫02∫0214xydxdy=14∫02y(∫02xdx)dy=14∫02yx22∣02dy=14∫02y(222−022)dy=14∫022ydy=12∫02ydy=12y22∣02=12(222−022)=122=1\begin{aligned} E[XY] &= \int\int f(x,y) xy dxdy \\ &= \int_{0}^{2}\int_{0}^2 \frac{1}{4} xy dxdy \\ &= \frac{1}{4} \int_{0}^{2} y (\int_{0}^{2} x dx) dy \\ &= \frac{1}{4} \int_{0}^{2} y \frac{x^{2}}{2} \vert_{0}^{2} dy \\ &= \frac{1}{4} \int_{0}^{2} y (\frac{2^{2}}{2} - \frac{0^{2}}{2}) dy \\ &= \frac{1}{4} \int_{0}^{2} 2 y dy \\ &= \frac{1}{2} \int_{0}^{2} y dy \\ &= \frac{1}{2} \frac{y^{2}}{2} \vert_{0}^{2} \\ &= \frac{1}{2} (\frac{2^{2}}{2} - \frac{0^{2}}{2}) \\ &= \frac{1}{2} 2 \\ &= 1 \end{aligned}

but

E[X]=E[Y]=∫02x12dx=1E[X] = E[Y] = \displaystyle\int_{0}^{2} x \displaystyle\frac{1}{2} dx = 1

so

E[XY]=E[X]E[Y]E[XY] = E[X] E[Y]

Independence and the Covariance​

If XX and YY are independent then Cov(X,Y)=0Cov(X,Y)=0

In fact, if XX and YY are independent then Cov(h(X),g(Y))=0Cov(h(X),g(Y))=0 for any functions gg and hh in which expected values exist.

The Moment Generating Function​

If XX is a random variable we define the moment generating function when tt exists as: M(t):=E[etX]M(t):=E[e^{tX}].

Examples​

Example

If X∼Bin(n,p)X\sim Bin(n,p) then M(t)=∑x=0netxp(x)=∑x=0netx(nx)p⋅(1−p)n−xM(t)=\displaystyle\sum_{x=0}^{n} e^{tx}p(x) = \displaystyle\sum_{x=0}^{n} e^{tx} \displaystyle\binom{n}{x}p\cdot (1-p)^{n-x}

Moments and the Moment Generating Function​

If MX(t)M_{X}(t) is the moment generating function (mgf) of XX, then MX(n)(0)=E[Xn]M_{X}^{(n)}(0)=E[X^n].

Details​

Observe that M(t)=E[etX]=E[1+X+(tX)22!+(tX)33!+… ]M(t)=E[e^{tX}]=E[1+X+\displaystyle\frac{(tX)^2}{2!}+\displaystyle\frac{(tX)^3}{3!}+\dots] since ea=1+a+a22!+a33!+…e^a=1+a+\displaystyle\frac{a^2}{2!}+\displaystyle\frac{a^3}{3!}+\dots. +If the random variable e∣tX∣e^{|tX|} has a finite expected value then we can switch the sum and the expected valued to obtain:

M(t)=E[∑n=0∞(tX)nn!]=∑n=0∞E[(tX)n]n!=∑n=0∞tnE[Xn]n!M(t)=E\left[\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac{(tX)^n}{n!}\right]=\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac{E[(tX)^n]}{n!}=\displaystyle\sum_{n=0}^{\infty}t^n\displaystyle\frac{E[X^n]}{n!}

This implies that the nthn^{th} derivative of M(t)M(t) evaluated at t=0t=0 is exactly E[Xn]E[X^n].

The Moment Generating Function of a Sum of Random Variables​

MX+Y(t)=MX(t)⋅MY(t)M_{X+Y}(t)=M_{X}(t)\cdot M_{Y}(t) if XX and YY are independent.

Details​

Let XX and YY be independent random vaiables, then

MX+Y(t)=E[eXt+Yt]=E[eXteXt]=E[eXt]E[eXt]=MX(t)MY(t)M_{X+Y}(t)=E[e^{Xt+Yt}]=E[e^{Xt}e^{Xt}]=E[e^{Xt}]E[e^{Xt}]=M_{X}(t)M_{Y}(t)

Uniqueness of the Moment Generating Function​

Moment generating functions (m.g.f.) uniquely determine the probability distribution function for random variables. +Thus, if two random variables have the same moment-generating function, then they must also have the same distribution.

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Multivariate Calculus

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Multivariate Calculus

Vector Functions of Several Variables​

A vector-valued function of several variables is a function

f:Rm→Rnf: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}

i.e. a function of mm -dimensional vectors, which returns nn dimensional vectors.

Examples​

Example

A real valued function of many variables: f:R3→Rf: \mathbb{R}^3\to\mathbb{R}, f(x1,x2,x3)=2x1+3x2+4x3f(x_1,x_2,x_3)=2x_1+3x_2+4x_3.

Note

ff is linear and f(x)=Axf(x)=Ax where

x=(x1x2x3)x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}

and

A=[234]A = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}
Example

Let

f:R2→R2f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}

where

f(x1,x2)=(x1+x2x1−x2)f(x_1,x_2) = \left( \begin{array}{c} x_1 + x_2 \\ x_1 - x_2 \end{array} \right)
Note

Note that f(x)=Axf(x)=Ax, where

A=[111−1]A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix}
Example

Let

f:R3→R4f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}

be defined by

f(x)=(x1+x2x1−x3y−zx1+x2+x3)f(x) = \left( \begin{array}{c} x_1 + x_2 \\ x_1 - x_3 \\ y - z \\ x_1 + x_2 + x_3 \end{array} \right)
Note

f(x)=Axf(x) = Ax

where

A=[11010−101−1111]A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix}
Example

These multi-dimensional functions do not have to be linear, for example the function f:R2→R2f:\mathbb{R}^2\to\mathbb{R}^2

f(x)=(x1⋅x2x12+x22)f(x) = \left( \begin{array}{c} x_1 \cdot x_2 \\ x_1^{2} +x_2^{2} \end{array} \right)

is obviously not linear.

The Gradient​

Suppose the real valued function f:Rm→Rf:\mathbb{R}^m \rightarrow \mathbb{R} is differentiable in each coordinate. +Then the gradient of ff, denoted ∇f\nabla f is given by

∇f(x)=(∂f∂x1,…,∂f∂x1)\nabla f(x)=\begin{pmatrix}\displaystyle\frac{\partial f}{\partial x_1},&\dots &,\displaystyle\frac{\partial f}{\partial x_1}\end{pmatrix}

Details​

Definition

Suppose the real valued function f:Rm→Rf:\mathbb{R}^m \rightarrow \mathbb{R} is differentiable in each coordinate. +Then the gradient of ff, denoted ∇f\nabla f is given by

∇f(x)=(∂f∂x1,…,∂f∂x1)\nabla f(x)= \begin{pmatrix} \displaystyle\frac{\partial f}{\partial x_1},&\dots &,\displaystyle\frac{\partial f}{\partial x_1}\end{pmatrix}

where each partial derivative ∂f∂xi\displaystyle\frac{\partial f}{\partial x_i} is computed by differentiating ff with respect to that variable, regarding the others as fixed.

Examples​

Example

Let

f(x‾)=x2+y2+2xyf(\underline{x})= x^2+y^2+2xy

Then the partial derivatives of ff are

∂f∂x=2x+2y\displaystyle\frac{\partial f}{\partial x}=2x+2y

and

∂f∂y=2y+2x\displaystyle\frac{\partial f}{\partial y}=2y+2x

and the gradient of ff is therefore

∇f=(2x+2y,2y+2x)\nabla f =\begin{pmatrix}2x+2y, & 2y+2x\end{pmatrix}

Example

Let

f(x‾)=x1−x2f(\underline{x})=x_1-x_2

The gradient of ff is

∇f=(1,−1)\nabla f= \begin{pmatrix}1, & -1\end{pmatrix}

The Jacobian​

Now consider a function f:Rm→Rnf:\mathbb{R}^m\to\mathbb{R}^n. +Write fif_i for the ithi^{th} coordinate of ff, so we can write f(x)=(f1(x),f2(x),…,fn(x))f(x)=(f_1(x),f_2(x),\ldots,f_n(x)), where x∈Rmx\in\mathbb{R}^m. +If each coordinate function fif_i is differentiable in each variable we can form the Jacobian matrix of ff:

(∇f1⋮∇fn)\begin{pmatrix} \nabla f_1 \\ \vdots \\ \nabla f_n \end{pmatrix}

Details​

Now consider a function f:Rm→Rnf:\mathbb{R}^m\to\mathbb{R}^n. +Write fif_i for the ithi^{th} coordinate of ff, so we can write f(x)=(f1(x),f2(x),…,fn(x))f(x)=(f_1(x),f_2(x),\ldots,f_n(x)), where x∈Rmx\in\mathbb{R}^m. +If each coordinate function fif_i is differentiable in each variable we can form the Jacobian matrix of ff:

(∇f1⋮∇fn)\begin{pmatrix} \nabla f_1 \\ \vdots \\ \nabla f_n \end{pmatrix}

In this matrix, the element in the ithi^{th} row and jthj^{th} column is ∂fi∂xj\displaystyle\frac{\partial f_i}{\partial x_j}.

Examples​

Example

For the function

f(x,y)=(x2+yxyx)=(f1(x,y)f2(x,y)f3(x,y))f(x,y) = \begin{pmatrix} x^2 +y \\ x y \\ x \end{pmatrix} = \begin{pmatrix} f_1(x,y) \\ f_2(x,y) \\ f_3(x,y) \end{pmatrix}

the Jacobian matrix of ff is the matrix

J=[∇f1∇f2∇f3]=[2x2yyx10]J = \begin{bmatrix} \nabla f_1 \\ \nabla f_2 \\ \nabla f_3 \end{bmatrix} = \begin{bmatrix} 2x & 2y \\ y & x \\ 1 & 0 \end{bmatrix}

Univariate Integration By Substitution​

If ff is a continuous function and gg is strictly increasing and differentiable then,

∫g(a)g(b)f(x)dx=∫abf(g(t))g′(t)dt\displaystyle\int_{g(a)}^{g(b)} f(x)dx = \displaystyle\int_a^b f(g(t))g^\prime (t)dt

Details​

If ff is a continuous function and gg is strictly increasing and differentiable then,

∫g(a)g(b)f(x)dx=∫abf(g(t))g′(t)dt\displaystyle\int_{g(a)}^{g(b)} f(x)dx = \displaystyle\int_a^b f(g(t))g^\prime (t)dt

It follows that if XX is a continuous random variable with density ff

and Y=h(X)Y = h(X) is a function of XX that has the inverse g=h−1g=h^{-1}, so X=g(Y)X = g(Y), then the density of YY is given by,

fY(y)=f(g(y))g′(y)f_Y(y) = f (g(y)) g^\prime (y)

This is a consequence of

P[Y≤b]=P[g(Y)≤g(b)]=P[X≤g(b)]=∫−∞g(b)f(x)dx=∫−∞bf(g(y))g′(y)dyP [Y \leq b] = P [g(Y) \leq g(b)] = P [X \leq g(b)] = \displaystyle\int_{- \infty} ^{g(b)}f(x)dx = \displaystyle\int_{- \infty} ^b f (g(y))g^\prime (y)dy

Multivariate Integration By Substitution​

Suppose ff is a continuous function f:Rn→Rf: \mathbb{R}^n \rightarrow \mathbb{R} and g:Rn→Rng: \mathbb{R}^n \rightarrow \mathbb{R}^n is a one-to-one function with continuous partial derivatives. +Then if U⊆RnU \subseteq \mathbb{R}^n is a subset,

∫g(U)f(x)dx=∫U(g(y))∣J∣dy\displaystyle\int_{g(U)} f(\mathbf {x})d\mathbf {x} = \displaystyle\int_{U}({g}(\mathbf {y}))|J|d\mathbf {y}

where JJ is the Jacobian matrix and ∣J∣|J| is the absolute value of it's determinant.

J=∣[∂g1∂y1∂g1∂y2⋯∂g1∂yn⋮⋮⋯⋮∂gn∂y1∂gn∂y2⋯∂gn∂yn]∣=∣[∇g1⋮∇gn]∣J = \left| \begin{bmatrix} \displaystyle\frac{\partial g_1}{\partial y_1} & \displaystyle\frac{\partial g_1}{\partial y_2} & \cdots &\displaystyle\frac{\partial g_1}{\partial y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \displaystyle\frac{\partial g_n}{\partial y_1} & \displaystyle\frac{\partial g_n}{\partial y_2} & \cdots & \displaystyle\frac{\partial g_n}{\partial y_n} \end{bmatrix}\right| = \left|\begin{bmatrix} \nabla g_1 \\ \vdots \\ \nabla g_n \end{bmatrix} \right|

Details​

Suppose ff is a continuous function f:Rn→Rf: \mathbb{R}^n \rightarrow \mathbb{R} and g:Rn→Rng: \mathbb{R}^n \rightarrow \mathbb{R}^n is a one-to-one function with continuous partial derivatives. +Then if U⊆RnU \subseteq \mathbb{R}^n is a subset,

∫g(U)f(x)dx=∫U(g(y))∣J∣dy\displaystyle\int_{g(U)} f(\mathbf {x})d\mathbf {x} = \displaystyle\int_{U}({g}(\mathbf {y}))|J|d\mathbf {y}

where JJ is the Jacobian determinant and |J| is its absolute value.

J=∣[∂g1∂y1∂g1∂y2⋯∂g1∂yn⋮⋮⋯⋮∂gn∂y1∂gn∂y2⋯∂gn∂yn]∣=∣[∇g1⋮∇gn]∣J = \left| \begin{bmatrix} \displaystyle\frac{\partial g_1}{\partial y_1} & \displaystyle\frac{\partial g_1}{\partial y_2} & \cdots &\displaystyle\frac{\partial g_1}{\partial y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \displaystyle\frac{\partial g_n}{\partial y_1} & \displaystyle\frac{\partial g_n}{\partial y_2} & \cdots & \displaystyle\frac{\partial g_n}{\partial y_n} \end{bmatrix}\right| = \left|\begin{bmatrix} \nabla g_1 \\ \vdots \\ \nabla g_n \end{bmatrix} \right|

Similar calculations as in 28.4 give us that if XX is a continuous multivariate random variable, X=(X1,…,Xn)′X = (X_1, \ldots, X_n)^\prime with density ff and Y=h(X)\mathbf{Y} = \mathbf{h} (\mathbf{X}), where h\mathbf{h} is one-to-one with inverse g=h−1\mathbf g= \mathbf{h}^{-1}. +So, X=g(Y)\mathbf{X} = g(\mathbf{Y}), then the density of Y\mathbf{Y} is given by;

fY(y)=f(g(y))∣J∣f_Y(\mathbf y) = f (g(\mathbf y)) |J|

Examples​

Example

If Y=AX\mathbf{Y} = A \mathbf X where AA is an n×nn \times n matrix with det⁡(A)≠0\det(A)\neq0 and X=(X1,…,Xn)′X = (X_1, \ldots, X_n)^\prime are independent and identically distributed random variables, then we have the following results.

The joint density of X1⋯XnX_1 \cdots X_n is the product of the individual (marginal) densities,

fX(x)=f(x1)f(x2)⋯f(xn)f_X(\mathbf x)= f(x_1) f(x_2) \cdots f(x_n)

The matrix of partial derivatives corresponds to ∂g∂y\displaystyle\frac{\partial g}{\partial y} where X=g(Y)\mathbf X = \mathbf g(\mathbf{Y}), i.e. these are the derivatives of the transformation: X=g(Y)=A−1Y\mathbf X = g (\mathbf{Y}) = A^{-1}\mathbf{Y}, or X=BY\mathbf X = B \mathbf{Y} where B=A−1B = A^{-1}

But if X=BY\mathbf X = B \mathbf{Y}, then

Xi=bi1y1+bi2y2+⋯bijyj⋯binynX_i = b_{i1}y_1 + b_{i2}y_2 + \cdots b_{ij}y_j\cdots b_{in}y_n

So, ∂xi∂yj=bij\displaystyle\frac{\partial x_i}{\partial y_j} = b_{ij} and thus,

J=∣∂dx∂dy∣=∣B∣=∣A−1∣=1∣A∣J =\left|\displaystyle\frac{\partial d\mathbf x}{\partial d\mathbf y}\right| = |B| = |A^{-1}| = \displaystyle\frac {1}{|A|}

The density of Y\mathbf{Y} is therefore;

fY(y)=fX(g(y))∣J∣=fX(A−1y)∣A−1∣f_Y(\mathbf{y}) = f_X(g(\mathbf{y})) |J| = f_X(A^{-1}\mathbf{y}) |A^{-1}|

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Notes and Examples, The Linear Model

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Notes and Examples: The Linear Model

Simple Linear Regression In R​

To test the effect of one variable on another, simple linear regression may be applied. +The fitted model may be expressed as y=α+β^xy=\alpha + \hat{\beta} x, where α\alpha is a constant, β^\hat{\beta} is the estimated coefficient, and xx is the explanatory variable.

Fig. 40

Figure: Example taken from R of a fitted model using linear regression.

Details​

Below is the linear regression output using the R's data set car. +Notice that the output from the model may be divided into two main categories:

  1. output that assesses the model as a whole, and

  2. output that relates to the estimated coefficients for the model

Call: lm(formula = dist ~ speed, data = cars)

Residuals: Min 1Q Median 3Q Max -29.069 -9.525 -2.272 9.215 43.201

Coefficients: Estimate Std.
Error t value Pr(>|t|) (Intercept) -17.5791 6.7584 -2.601 0.0123 * speed 3.9324 0.4155 9.464 1.49e-12 *** ---

Residual standard error: 15.38 on 48 degrees of freedom Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438 F-statistic: 89.57 on 1 and 48 DF, p-value: 1.490e-12

Notice that there are four different sets of output (Call, Residuals, Coefficients, and Results) for both the constant α\alpha and the estimated coefficient β^\hat{\beta} speed variable.

The estimated coefficients describe the change in the dependent variable when there is a single unit increase in the explanatory variable given that everything else is held constant.

The standard error is a measure of accuracy and is used to construct the confidence interval. +Confidence intervals provide a range of values for which there is a set level of confidence that the true population mean will be within the given range. +For example, if the CI is set at 95% percent then the probability of observing a value outside the given CI range is less than 0.05

The pp-value is represented as a percentage. +Specifically, the pp-value indicates the percentage of time, given that your null hypothesis is true, that you would find an outcome at least as extreme as the observed value. +If your calculated pp-value is 0.020.02 then 2% of the time you'd observe a mean at least as large as your observed

In the overall model assessment the R-squared is the explained variance over the total variance. +Generally, a higher R2R^2 is better but data with very little variance makes it easy to achieve a higher R2R^2, which is why the adjusted R2R^2 is presented

Lastly, the FF-statistic is given. +Since the tt-Statistic is not appropriate to compare two or more coefficients, the FF-statistic must be applied. +The basic methodology is that it compares a restricted model where the coefficients have been set to a certain fixed level to a model which is unrestricted. +The most common is the sum of squared residuals FF-test.

Multiple Linear Regression​

The One-way Model​

The one-way ANOVA model is of the form:

Yij=μi+ϵijY_{ij}=\mu_i+\epsilon_{ij}

or

Yij=μ+αi+ϵijY_{ij}=\mu+\alpha_i+\epsilon_{ij}

Details​

The one-way ANOVA model is of the form:

Yij=μi+ϵijY_{ij}=\mu_i+\epsilon_{ij}

where YijY_{ij} is observation jj in treatment group ii and μi\mu_i are the parameters of the model and are means of treatment group ii. +The ϵij\epsilon_{ij} are independent and follow a normal distribution with mean zero and constant variance σ2\sigma^2 often written as ϵ∼N(0,σ2)\epsilon\sim N(0,\sigma^2).

The ANOVA model can also be written in the form:

Yij=μ+αi+ϵijY_{ij}=\mu+\alpha_i+\epsilon_{ij}

where μ\mu is the overall mean of all treatment groups and αi\alpha_i is the deviation of mean of treatment group ii from the overall mean. +The ϵij\epsilon_{ij} follow a normal distribution as before

The expected value of YijY_{ij} is μi\mu_i as the expected value of the errors is zero, often written as E[Yij]=μiE[Y_{ij}]=\mu_i.

Examples​

Example

In the rat diet experiment the model would be of the form:

yij=μi+ϵijy_{ij}=\mu_i+\epsilon_{ij}

where yijy_{ij} is the weight gain for rat jj in diet group ii, μi\mu_i would be the mean weight gain in diet group ii and ϵij\epsilon_{ij} would be the deviation of rat jj from the mean of its diet group.

Random Effects In the One-way Layout​

The simplest random effects model is the one-way layout, commonly written in the form

yij=μ+αi+ϵij,y_{ij}=\mu + \alpha_i + \epsilon_{ij},

where j=1,…,Jj =1,\ldots,J and i=1,…,Ii =1,\ldots,I.

Normally one also assumes ϵij∼N(0,σA2)\epsilon_{ij}\sim N(0,\sigma_A^2), αi∼N(0,σA2)\alpha_i \sim N (0,\sigma_A^2), and that all these random variables are independent.

Note that we have stopped making a distinction in notation between random variables and measurements (the yy -values are just random variables when distributions occur).

Details​

Note that this is considerably different from the fixed effect model.

Since the factor has changed to a random variable with an expected value of zero, the expected value of all the yy is the same:

Eyij=μEy_{ij}=\mu

The variance of yy now has two components:

Vyij=σA2+σ2Vy_{ij}=\sigma^2_A + \sigma^2

In addition we have a covariance structure between the measurements and this needs to be looked at in some detail.. +First, the general case of a covariance between two general yijy_{ij} and yi′j′y_{i'j'}, where the indices may or may not be the same:

Cov(yij,yi′j′)=Cov(αi+ϵij,αi′+ϵi′j′)=E[(αi+ϵij)(αi′+ϵi′j′)]=E[αiαi′]+E[ϵijαi′]+E[αiϵi′j′]+E[ϵijϵi′j′]\begin{aligned} Cov(y_{ij},y_{i'j'}) &= Cov(\alpha_i+\epsilon_{ij}, \alpha_{i'}+ \epsilon_{i'j'}) \\ &= E[(\alpha_i+\epsilon_{ij})(\alpha_{i'}+\epsilon_{i'j'})] \\ &= E[\alpha_i\alpha_{i'}] + E[\epsilon{ij}\alpha_{i'}]+ E[\alpha_i\epsilon_{i'j'}] + E[\epsilon_{ij}\epsilon_{i'j'}] \end{aligned}
Note

Recall that E[UW]=E[U]E[W]E[UW]=E[U]E[W] if U,WU,W are independent

So

E[ϵijαi′]=E[αiϵi′j′]=EαiEϵi′j′=0E[\epsilon_{ij}\alpha_{i'}]=E[\alpha_i\epsilon_{i'j'}] = E\alpha_iE\epsilon_{i'j'}=0

Further,

E[ϵijϵi′j′]={σ2if i=i′,j=j′0otherwiseE[\epsilon_{ij}\epsilon{i'j'}] = \begin{cases} \sigma^2 & \text{if } i=i', j=j' \\ 0 & \text{otherwise} \end{cases}

and

E[αiαi′]={σA2if i=i′0if i≠i′E[\alpha_{i}\alpha{i'}] = \begin{cases} \sigma^2_A &\text{if } i=i' \\ 0 &\text{if }i \neq i' \end{cases}

so

Cov(yij,yi′j′)={σA2+σ2if i=i′,j=j′σA′if i=i′,j≠j′0otherwiseCov(y_{ij},y_{i'j'}) = \begin{cases} \sigma_A^2+\sigma^2 & \text{if } i=i', j=j' \\ \sigma'_A & \text{if } i=i', j \neq j' \\ 0 & \text{otherwise} \end{cases}

It follows that the correlation between measurements yijy_{ij} and yij′y_{ij'} (within the same group) are

Cor(yij,yij′)=Cov(yij,yij′)Var[yij]Var[yij′]=σA2(σA2+σ2)2=σA2σA2+σ2\begin{aligned} Cor(y_{ij},y_{ij'}) &= \displaystyle\frac{Cov(y_{ij},y_{ij'})}{\sqrt{Var[y_{ij}]Var[y_{ij'}]}} \\ &= \displaystyle\frac{\sigma_A^2}{\sqrt{(\sigma_A^2 + \sigma^2)^2}} \\ &= \displaystyle\frac{\sigma_A^2}{\sigma_A^2 + \sigma^2} \end{aligned}

This is the intra-class correlation.

Linear Mixed Effects Models (lmm)​

The simplest mixed effects model is

yij=μ+αi+βj+ϵijy_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij}

where μ,α1,α2,…,αi\mu, \alpha_1, \alpha_2, \ldots, \alpha_i are unknown constants,

βj∼N(0,σβ2)\beta_j \sim N(0,\sigma^2_\beta)

ϵij∼N(0,σ2)\epsilon_{ij} \sim N(0,\sigma^2)

(βj\beta_j and ϵij\epsilon_{ij} independent).

Details​

The μ\mu and αi\alpha_i are the fixed effects and βj\beta_j is the random effects. +Recall that in the simple one-way layout with yij=μ+αi+ϵijy_{ij} = \mu + \alpha_i + \epsilon_{ij}, we can write the model in matrix form y‾=Xβ‾+ϵ‾\underline{y} = X \underline{\beta} + \underline{\epsilon} where β‾=(μ,α1,…,αI)′\underline{\beta} = (\mu, \alpha_1, \ldots, \alpha_I)' and XX is appropriately chosen.

The same applies to the simplest random effects model yij=μ+βj+ϵijy_{ij}= \mu + \beta_j+ \epsilon_{ij} where we can write y‾=μ⋅1‾+ZU‾+ϵ‾\underline{y} = \mu \cdot \underline{1}+ Z \underline{U} + \underline{\epsilon} where 1‾=(1,1,…,1)′\underline{1}=(1,1, \ldots, 1)', U‾=(β1,…,βJ)′\underline{U} = ( \beta_1, \ldots, \beta_J )'.

In general, we write the mixed effects models in matrix form with y‾=Xβ‾+ZU‾+ϵ‾\underline{y} = X \underline{\beta} + Z \underline{U} + \underline {\epsilon}, where β‾\underline{\beta} contains the fixed effects and U‾\underline{U} contains the random effects.

Examples​

Example

  1. yi=β1+β2xi+ϵiy_i = \beta_1 + \beta_2 x_i + \epsilon_i (SLR)

  2. yij=μ+αi+βixij+ϵijy_{ij} = \mu + \alpha_i + \beta_i x_{ij} + \epsilon_{ij} only fixed effects (ANCOVA)

  3. yijk=μ+αi+bj+ϵijky_{ijk} = \mu + \alpha_i + b_j + \epsilon_{ijk} where αi\alpha_i are fixed but bjb_j are random.

  4. yijk=μ+αi+bjxij+ϵijky_{ijk} = \mu + \alpha_i + b_j x_{ij} + \epsilon_{ijk} where αi\alpha_i are fixed but bjb_j are random slopes.

Maximum Likelihood Estimation In Lmm​

The likelihood function for the unknown parameters L(β,σA2,σ2)L(\boldsymbol{\beta},\sigma^2_A, \sigma^2) is

1(2π)n/2∣Σy∣n/2e−1/2(y−Xβ)′Σy−1(y−Xβ)\displaystyle\frac{1}{(2\pi)^{n/2} \left| \boldsymbol{\Sigma}_y \right| ^{n/2}} e^{-1/2 (\mathbf{y}-X\boldsymbol{\beta})' \boldsymbol{\Sigma}^{-1}_y (\mathbf{y}-X\boldsymbol{\beta})}

where Σy=σA2ZZ′+σ2I\boldsymbol{\Sigma}_y = \sigma^2_A Z Z' + \sigma^2 I

Maximising LL over β,σA2,σ2\boldsymbol{\beta},\sigma^2_A, \sigma^2 gives the variance components and the fixed effects. +May also need u^\mathbf{\hat{u}}, this is normally done using BLUP.

Details​

Recall that if WW is a random variable vector with EW=μEW = \mu and VW=ΣVW= \boldsymbol{\Sigma} then

E[AW]=AμE[AW] = A\mathbf{\mu}

Var[AW]=AΣA′Var[AW]= A \boldsymbol{\Sigma} A'

In particular, if W∼N(μ,Σ)W \sim N(\mu, \boldsymbol{\Sigma}) then AW∼N(Aμ,AΣA′)AW \sim N(A\mu, A \boldsymbol{\Sigma} A')

Now consider the lmm with

y=Xβ+Zu+ϵy = X \boldsymbol{\beta} + Zu + \boldsymbol{\epsilon}

where

u=(u1,…,um)′u = (u_1, \ldots, u_m)'

ϵ=(ϵ1,…,ϵm)′\boldsymbol{\epsilon} = (\epsilon_1, \ldots, \epsilon_m)'

and the random variables Ui∼N(0,σA2)U_i \sim N(0, \sigma^2_A), ϵi∼N(0,σ2)\epsilon_i \sim N(0, \sigma^2) are all independent so that u∼N(0,σA2I)u \sim N(0, \sigma^2_A I) and ϵ∼N(0,σ2I)\boldsymbol{\epsilon} \sim N(\mathbf{0}, \sigma^2 I).

Then Ey=XβEy = X\boldsymbol{\beta} and

Vy=Σy=Var[Zu+Var[ϵ]]=Z(σA2I)Z′+σ2I=σA2ZZ′+σ2I\begin{aligned} Vy &= \boldsymbol{\Sigma}_y \\ &= Var[Zu+Var[\boldsymbol{\epsilon}]] \\ &= Z(\sigma^2_A I) Z' + \sigma^2 I \\ &= \sigma^2_A Z Z' + \sigma^2 I \end{aligned}

and hence y∼N(Xβ,σA2ZZ′+σ2I)y \sim N(X\boldsymbol{\beta},\sigma^2_A Z Z' + \sigma^2 I )

Therefore the likelihood function for the unknown parameters L(β,σA2,σ2)L(\boldsymbol{\beta},\sigma^2_A, \sigma^2) is

=1(2π)n/2∣Σy∣n/2e−1/2(y−Xβ)′Σy−1(y−Xβ)= \displaystyle\frac{1}{(2\pi)^{n/2} \left| \boldsymbol{\Sigma}_y \right| ^{n/2}} e^{-1/2 (\mathbf{y}-X\boldsymbol{\beta})' \boldsymbol{\Sigma}^{-1}_y (y-X\boldsymbol{\beta})}

where Σy=σA2ZZ′+σ2I\boldsymbol{\Sigma}_y = \sigma^2_A Z Z' + \sigma^2 I. +Maximizing LL over β,σA2,σ2\boldsymbol{\beta},\sigma^2_A, \sigma^2 gives the variance components and the fixed effects. +May also need u^\hat{u}, which is normally done using BLUP.

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Ranks and Determinants

The Rank of a Matrix​

The rank of an n×pn \times p matrix AA, denoted by rank(A)\text{rank}(A), is the largest number of columns of AA, which are not linearly dependent (i.e. the number of linearly independent columns).

Details​

Vectors a1,a2,…,ana_1, a_2, \ldots, a_n are said to be linearly dependent if there exist constants k1,…,knk_1, \ldots, k_n that are not all zero, such that

k1a1+k2a2+…+knan=0k_1 a_1 + k_2 a_2 + \ldots + k_n a_n = 0

Note that if such constants exist, then we can write one of the aa 's as a linear combination of the rest, e.g. if k1≠0k_1 \neq 0 then

a1=c1=−k2k1a2−…−k2k1ana_1=\mathbf{c_1} = -\displaystyle\frac{k_2}{k_1} a_2 - \ldots - \displaystyle\frac{k_2}{k_1} a_n

It can be shown that the rank of AA, is the same as the rank of A′A'

i.e. the maximum number of linearly independent rows of AA.

Note

Note that if rank(A)=p\text{rank}(A) = p, then the columns are linearly independent.

Examples​

Example

If

A=[1001]A = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]

then rank(A)\text{rank}(A) = 2, since

k1(10)+k2(01)=(00)k_1 \left( \begin{array}{cc} 1 \\ 0 \end{array} \right) + k_2 \left( \begin{array}{cc} 0 \\ 1 \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right)

if and only if

(k1k2)=(00)\left( \begin{array}{cc} k_1 \\ k_2 \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right)

so the columns are linearly independent.

Example

If

A=[101011000]A = \left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array} \right]

then rank(A)\text{rank}(A) = 2.

Example

If

A=[111010010]A = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array} \right]

then rank(A)\text{rank}(A) = 2. since

1(100)+0(011)+(−1)(100)=01 \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right) + 0 \left( \begin{array}{ccc} 0 \\ 1 \\ 1 \end{array} \right) + (-1) \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right) = 0

(and hence the rank cannot be more than 2) but

k1(100)+k2(011)k_1 \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right) + k_2 \left( \begin{array}{ccc} 0 \\ 1 \\ 1 \end{array} \right)

if and only if k1=k2=0k_1=k_2=0 (and hence the rank must be at least 2).

The Determinant​

Recall that for a 2×22 \times 2 matrix,

A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

the inverse of AA is

A−1=1ad−bc[2331]A^{-1} = \displaystyle\frac{1}{ad-bc} \begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix}

Details​

Definition

The number ad−bcad-bc is called the determinant of the 2×22 \times 2 matrix AA.

Definition

Now suppose AA is an n×nn \times n matrix. +An elementary product from the matrix is a product of nn terms based on taking exactly one term from each column of row xx. +Each such term can be written in the form a1j1⋅a2j2⋅a3j3⋅…⋅anjna_{1j_1} \cdot a_{2j_2} \cdot a_{3j_3} \cdot \ldots \cdot a_{nj_n} where j1,…,jnj_1, \ldots, j_n is a permutation of the integers 1,2,…,n1,2, \ldots, n. +Each permutation σ\sigma of the integers 1,2,…,n1,2,\ldots,n can be performed by repeatedly interchanging two numbers.

Definition

A signed elementary product is an elementary product with a positive sign if the number of interchanges in the permutation is even but negative otherwise.

The determinant of AA, det⁡(A)\det(A) or ∣A∣\vert A \vert, is the sum of all signed elementary products.

Examples​

Example
A=[a11a12a21a22]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

then

∣A∣=a11‾a22‾−a12‾a21‾\vert A \vert = a_{1\underline{1}} a_{2\underline{2}} - a_{1\underline{2}}a_{2\underline{1}}.

Example

If

A=[a11a12a13a21a22a23a31a32a33]A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}

Then ∣A∣\vert A \vert

= a11a22a33a_{11} a_{22} a_{33} This is the identity permutation and has positive sign

−a11a23a32-a_{11} a_{23} a_{32} This is the permutation that only interchanges 22 and 33

−a12a21a33-a_{12} a_{21} a_{33} Only one interchange

+a12a23a31+a_{12} a_{23} a_{31} Two interchanges

+a13a21a32+a_{13} a_{21} a_{32} Two interchanges

−a13a22a31-a_{13} a_{22} a_{31} Three interchanges

Example
A=[1110]A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}

∣A∣=−1\vert A \vert = -1

Example
A=[100020003]A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}

∣A∣=1⋅2⋅3=6\vert A \vert = 1 \cdot 2 \cdot 3 = 6

Example
A=[100020030]A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 3 & 0 \end{bmatrix}

∣A∣=0\vert A \vert = 0

Example
A=[100002030]A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 3 & 0 \end{bmatrix}

∣A∣=−6\vert A \vert = -6

Example
A=[2121]A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}

∣A∣=0\vert A \vert = 0

Example
A=[101011112]A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}

∣A∣=0\vert A \vert = 0

Ranks, Inverses and Determinants​

The following statements are true for an n×nn\times n matrix AA :

  • rank(A)=n\text{rank} (A)= n

  • det⁡(A)≠0\det(A)\neq 0

  • AA has an inverse

Details​

Suppose AA is an n×nn\times n matrix. +Then the following are truths:

  • rank(A)=n\text{rank} (A)= n

  • det⁡(A)≠0\det(A)\neq 0

  • AA has an inverse

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Some Notes on Matrices and Linear Operators

The Matrix As a Linear Operator​

Let AA be an m×nm\times n matrix. +The function

TA:Rn→Rm,TA(x‾)=Ax‾,T_A:\mathbb{R}^n\to\mathbb{R}^m, T_A(\underline{x}) = A\underline{x},

is linear, that is

TA(ax‾+by‾)=aTA(x‾)+bTA(y‾)T_A (a\underline{x} + b\underline{y}) = aT_A(\underline{x}) + bT_A(\underline{y})

if x‾,y‾∈Rn\underline{x}, \underline{y} \in \mathbb{R}^n and a,b∈Ra, b \in \mathbb{R}.

Examples​

Example

If

A=[12]A = \begin{bmatrix} 1 & 2 \end{bmatrix}

then TA(x‾)=x+2yT_A(\underline{x}) = x + 2y where x‾=(xy)∈R2\underline{x} = \displaystyle{x \choose y}\in \mathbb{R}^2

Example

If

A=[0110]A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

then

TA(xy)=[yx]T_A\displaystyle{x \choose y} = \begin{bmatrix} y \\ x \end{bmatrix}
Example

If

A=[023101]A = \begin{bmatrix} 0 & 2 & 3 \\ 1 & 0 & 1 \end{bmatrix}

then

TA(xyz)=[2y+3zx+z]T_A \left( \begin{array}{ccc} x \\ y \\ z \end{array} \right) = \begin{bmatrix} 2y + 3z \\ x + z \end{bmatrix}
Example

If

T(xy)=(x+y2x−3y)T \displaystyle{x \choose y } = \left( \begin{array}{cc} x + y \\ 2x - 3y \end{array} \right)

then T(x‾)=Ax‾T (\underline{x}) = A \underline{x} if we set

A=[112−3]A = \begin{bmatrix} 1 & 1 \\ 2 & -3 \end{bmatrix}

Inner Products and Norms​

Assuming xx and yy are vectors, then we define their inner product by

x⋅y=x1y1+x2y2+⋯+xnynx \cdot y = x_1y_1 + x_2y_2 + \cdots + x_ny_n

where

x=(x1⋮xn)x = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}

and

y=(y1⋮yn)y = \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix}

Details​

If xx, yy ∈Rn\in \mathbb{R}^n are arbitrary (column) vectors, then we define their inner product by

x⋅y=x1y1+x2y2+⋯+xnynx \cdot y = x_1y_1 + x_2y_2 + \cdots + x_ny_n

where

x=(x1⋮xn)x = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}

and

y=(y1⋮yn)y = \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix}
Note

Note that we can also view xx and yy as n×1n \times 1 matrices and we see that x⋅y=x′yx \cdot y = x^\prime y.

Definition

The normal length of a vector is defined by ∥x∥2=x⋅x\left \| x \right \|^2 = x \cdot x. +It may also be expressed as ∥x∥=x12+x22+⋯+xn2\left \| x \right \| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.

It is easy to see that for vectors a,ba, b and cc we have (a+b)⋅c=a⋅c+b⋅c(a+b)\cdot c=a\cdot c+ b\cdot c and a⋅b=b⋅aa\cdot b=b\cdot a.

Examples​

Two vectors xx and yy are said to be orthogonal if x⋅y=0x \cdot y = 0

Example

If

x=(34)x = \begin{pmatrix} 3 \\ 4 \end{pmatrix}

and

y=(21)y = \begin{pmatrix} 2 \\ 1 \end{pmatrix}

then

x⋅y=3⋅2+4⋅1=10x \cdot y = 3 \cdot 2 + 4 \cdot 1 = 10

and

∥x∥2=32+42=25\left \| x \right \|^2 = 3^2 + 4^2 = 25

so

∥x∥=5\left\| x \right \| = 5

Orthogonal Vectors​

Two vectors xx and yy are said to be orthogonal if x⋅y=0x\cdot y=0 denoted x⊥yx \perp y.

Details​

Definition

Two vectors xx and yy are said to be orthogonal if x⋅y=0x\cdot y=0 denoted x⊥yx \perp y.

If a,b∈Rna,b \in \mathbb{R}^n then

∥a+b∥2=a⋅a+2a⋅b+b⋅b\left\|a+b\right\|^2=a\cdot a+2a\cdot b+b\cdot b

so

∥a+b∥2=∥a∥2+∥b∥2+2a‾b‾\left\|a+b\right\|^2=\left\|a\right\|^2+\left\|b\right\|^2 + 2\underline{a}\underline{b}

Note

Note that if a⊥ba \perp b then ∥a+b∥2=∥a∥2+∥b∥2\left\|a+b\right\|^2=\left\|a\right\|^2+ \left\|b\right\|^2, which is Pythagoras' theorem in nn dimensions.

Linear Combinations of Independent Identicallly Distributed Variables​

Suppose X1,…,XnX_1,\dots,X_n are independent identically distributed random variables and have mean μ1,…,μn\mu_1, \dots, \mu_n and variance σ2\sigma^2 then the expected value of YY of the linear combination is:

Y=∑aiXiY=\displaystyle\sum a_i X_i

and if a1,…,ana_1,\dots,a_n are real constants then the mean is:

μY=∑aiμi\mu_Y = \displaystyle\sum a_i \mu_i

and the variance is:

σ2=∑ai2σi2\sigma^2 = \displaystyle\sum a^2_i \sigma^2_i

Examples​

Example

Consider two independent identically distributed random variables, Y1,Y2Y_1,Y_2 such that E[Y1]=E[Y2]=2E[Y_1]=E[Y_2]=2 and Var[Y1]=Var[Y2]=4Var[Y_1]=Var[Y_2]=4, and a specific linear combination of the two, W=Y1+3Y2W=Y_1+3Y_2. +We first obtain

E[W]=E[Y1+3Y2]=E[Y1]+3E[Y2]=2+3⋅2=2+6=8E[W]=E[Y_1+3Y_2]=E[Y_1]+3E[Y_2]=2+3\cdot 2=2+6=8

Similarly, we can first use independence to obtain

Var[W]=Var[Y1+3Y2]=Var[Y1]+Var[3Y2]Var[W]=Var[Y_1+3Y_2]=Var[Y_1]+Var[3Y_2]

and then (recall that Var[aY]=a2Var[Y]Var[aY]=a^2Var[Y] )

Var[Y1]+Var[3Y2]=Var[Y1]+32Var[Y2]=12⋅4+32⋅4=1⋅4+9⋅4=40Var[Y_1]+Var[3Y_2]=Var[Y_1]+3^2Var[Y_2]=1^2 \cdot 4+3^2\cdot 4= 1 \cdot 4 + 9 \cdot 4= 40

Normally, we just write this up in a simple sequence

Var[W]=Var[Y1+3Y2]=Var[Y1]+32Var[Y2]=12⋅4+32⋅4=1⋅4+9⋅4=40Var[W]=Var[Y_1+3Y_2]=Var[Y_1]+3^2Var[Y_2]=1^2 \cdot 4+3^2\cdot 4 = 1 \cdot 4 + 9 \cdot 4= 40

Covariance Between Linear Combinations of Independent Identically Distributed Random Variables​

Suppose Y1,…,YnY_1,\ldots,Y_n are independent identically distributed, each with mean μ\mu and variance σ2\sigma^2 and a,b∈Rna,b\in \mathbb{R}^n. +Writing

Y=(Y1⋮Yn)Y = \left( \begin{array}{ccc} Y_1 \\ \vdots \\ Y_n \end{array} \right)

consider the linear combination a′Ya'Y and b′Yb'Y.

Details​

The covariance between random variables UU and WW is defined by

Cov(U,W)=E[(U−μu)(W−μw)]Cov(U,W)= E[(U-\mu_u)(W-\mu_w)]

where μu=E[U]\mu_u=E[U] and μw=E[W]\mu_w=E[W]. +Now, let U=a′Y=∑YiaiU=a'Y=\displaystyle\sum Y_ia_i and W=b′Y=∑YibiW=b'Y=\displaystyle\sum Y_ib_i, where Y1,…,YnY_1,\ldots,Y_n are independent identically distributed with mean μ\mu and variance σ2\sigma^2, then we get

Cov(U,W)=E[(a′Y−Σaμ)(b′Y−Σbμ)]Cov(U,W)= E[(a'Y-\Sigma a_\mu)(b'Y-\Sigma b\mu)]

=E[(ΣaiYi−Σaiμ)(ΣbjYj−Σbjμ)]= E[(\Sigma a_iY_i -\Sigma a_i\mu)(\Sigma b_jY_j -\Sigma b_j\mu )]

and after some tedious (but basic) calculations we obtain

Cov(U,W)=σ2a⋅bCov(U,W)=\sigma^2a\cdot b

Examples​

Example

If Y1Y_1 and Y2Y_2 are independent identically distributed, then

Cov(Y1+Y2,Y1−Y2)=Cov((1,1)(Y1Y2),(1,−1)(Y1Y2))Cov(Y_1+Y_2, Y_1-Y_2) = Cov \left( (1,1) \begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix}, (1,-1) \begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix} \right)
=(1,1)(1−1)σ2=0= (1,1) \begin{pmatrix} 1 \\ -1 \end{pmatrix} \sigma^2 = 0

and in general, Cov(a‾′Y‾,b‾′Y‾)=0Cov(\underline{a}'\underline{Y}, \underline{b}'\underline{Y})=0 if a‾⊥b‾\underline{a}\bot \underline{b} and Y1,…,YnY_1,\ldots,Y_n are independent.

Random Vectors​

Y=(Y1,…,Yn)Y= (Y_1, \ldots, Y_n) is a random vector if Y1,…,YnY_1, \ldots, Y_n are random variables.

Details​

Definition

If E[Yi]=μiE[Y_i] = \mu_i then we typically write

E[Y]=(μ1⋮μn)=μE[Y] = \left( \begin{array}{ccc} \mu_1 \\ \vdots \\ \mu_n \end{array} \right) = \mu

If Cov(Yi,Yj)=σijCov(Y_i, Y_j) = \sigma_{ij} and Var[Yi]=σii=σi2Var[Y_i]=\sigma_{ii} = \sigma_i^2, then we define the matrix

Σ=(σij)\boldsymbol{\Sigma} = (\sigma_{ij})

containing the variances and covariances. +We call this matrix the covariance matrix of YY, typically denoted Var[Y]=ΣVar[Y] = \boldsymbol{\Sigma} or CoVar[Y]=ΣCoVar[Y] = \boldsymbol{\Sigma}.

Examples​

Example

If Yi,…,YnY_i, \ldots, Y_n are independent identically distributed, EYi=μEY_i = \mu, VYi=σ2VY_i = \sigma^2, a,b∈Rna,b\in\mathbb{R}^n and U=a′YU=a'Y, W=b′YW=b'Y, and

T=[UW]T = \begin{bmatrix} U \\ W \end{bmatrix}

then

ET=[ΣaiμΣbiμ]ET = \begin{bmatrix} \Sigma a_i \mu \\ \Sigma b_i \mu \end{bmatrix}
VT=Σ=σ2[Σai2ΣaibiΣaibiΣbi2]VT = \boldsymbol{\Sigma} = \sigma^2 \begin{bmatrix} \Sigma a_i^2 & \Sigma a_i b_i \\ \Sigma a_ib_i & \Sigma b_i^2 \end{bmatrix}
Example

If Y‾\underline{Y} is a random vector with mean μ\boldsymbol{\mu} and variance-covariance matrix Σ\boldsymbol{\Sigma}, then

E[a′Y]=a′μE[a'Y] = a'\mu

and

Var[a′Y]=a′ΣaVar[a'Y] = a' \boldsymbol{\Sigma} a

Transforming Random Vectors​

Suppose

Y=(Y1⋮Yn)\mathbf{Y} = \left( \begin{array}{c} Y_1 \\ \vdots \\ Y_n \end{array} \right)

is a random vector with E[Y]=μE[\mathbf{Y}] = \mu and Var[Y]=ΣVar[\mathbf{Y}] = \boldsymbol{\Sigma} where the variance-covariance matrix

Σ=σ2I\boldsymbol{\Sigma} = \sigma^2 I

Details​

Note that if Y1,…,YnY_1, \ldots, Y_n are independent with common variance σ2\sigma^2 then

Σ=[σ12σ12σ13…σ1nσ21σ22σ23…σ2nσ31σ32σ32…σ3n⋮⋮⋮⋱⋮σn1σn2σn3…σn2]=[σ120……00σ22⋱0⋮⋮⋱σ32⋱⋮⋮0⋱⋱00……0σn2]=σ2[10……001⋱0⋮⋮⋱1⋱⋮⋮0⋱⋱00……01]=σ2I\boldsymbol{\Sigma} = \left[ \begin{array}{ccccc} \sigma_{1}^{2} & \sigma_{12} & \sigma_{13} & \ldots & \sigma_{1n} \\ \sigma_{21} & \sigma_2^{2} & \sigma_{23} & \ldots & \sigma_{2n} \\ \sigma_{31} &\sigma_{32} &\sigma_3^{2} & \ldots & \sigma_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \sigma_{n1} & \sigma_{n2} & \sigma_{n3} & \ldots & \sigma_n^{2} \end{array} \right] = \left[ \begin{array}{ccccc} \sigma_{1}^{2} & 0 & \ldots & \ldots & 0 \\ 0 & \sigma_2^{2} & \ddots & 0 & \vdots \\ \vdots & \ddots &\sigma_3^{2} & \ddots & \vdots \\ \vdots & 0 & \ddots & \ddots & 0 \\ 0 & \ldots & \ldots & 0 & \sigma_n^{2} \end{array} \right] = \sigma^2 \left[ \begin{array}{ccccc} 1 & 0 & \ldots & \ldots & 0 \\ 0 & 1 & \ddots & 0 & \vdots \\ \vdots & \ddots & 1 & \ddots & \vdots \\ \vdots & 0 & \ddots & \ddots & 0 \\ 0 & \ldots & \ldots & 0 & 1 \end{array} \right] = \sigma^2 I

If AA is an m×nm \times n matrix, then

E[AY]=AμE[A\mathbf{Y}] = A \mathbf{\mu}

and

Var[AY]=AΣA′Var[A\mathbf{Y}] = A \boldsymbol{\Sigma} A'

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Some Regression Topics

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Some Regression Topics

Poisson Regression​

Data yiy_i are from a Poisson distribution with mean μi\mu_i and ln⁡μi=β1+β2xi\ln{\mu_i}=\beta_1+\beta_2 x_i. +A likelihood function can be written and the parameters can be estimated using maximum likelihood.

The Generalized Linear Model (GLM)​

Data yiy_i are from a distribution within the exponential family, with mean μi\mu_i and g(μi)=xi′βg(\mu_i)=\textbf{x}'_i\boldsymbol{\beta} for some link function, gg. +A likelihood function can now be written and the parameters can be estimated using maximum likelihood.

Details​

Data yiy_i are from a distribution within the exponential family, with mean μi\mu_i and g(μi)=xi′βg(\mu_i)=\textbf{x}'_i\boldsymbol{\beta} for some link function, gg.

The exponential family includes distributions such as the Gaussian, binomial, Poisson, and gamma (and thus exponential and chi-squared)

The link functions are typically

  • identity (with the Gaussian)

  • log (with the Poisson and the gamma)

  • logistic (with the binomial)

A likelihood function can be set up for each of these models and the parameters can be estimated using maximum likelihood.

The glm package in R has options to estimate parameters in these models.

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The Gamma Distribution

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The Gamma Distribution

The Gamma Distribution​

If a random variable XX has the density

f(x)=xα−1e−xβΓ(α)βαf(x) = \displaystyle\frac{x^{\alpha-1} e^{\displaystyle\frac{-x} {\beta}}} {\Gamma(\alpha) \beta^{\alpha}}

where x>0x>0 for some constants α\alpha, β>0\beta>0, then XX is said to have a gamma distribution.

Details​

The function Γ\Gamma is basically chosen so that ff integrates to one, i.e.

Γ(α)=∫0∞tα−1e−tdt\Gamma(\alpha) = \displaystyle\int_0^\infty t^{\alpha-1} e^{-t}dt

It is not too hard to see that Γ(n)=(n−1)!\Gamma(n)=(n-1)! if n∈Nn \in \mathbb{N}. +Also, Γ(α+1)=αΓ(α)\Gamma(\alpha + 1) = \alpha \Gamma(\alpha) for all α>0\alpha > 0.

The Mean, Variance and Mgf of the Gamma Distribution​

Suppose X∼G(α,β)X \sim G (\alpha, \beta) i.e. XX has density

f(x)=xα−1e−x/βΓ(α)βα,x>0f(x) = \displaystyle\frac{x^{\alpha -1} e^{-x/\beta}} {\Gamma (\alpha) \beta^{\alpha}}, x > 0

Then,

E[X]=αβE[X] = \alpha\beta

M(t)=(1−βt)−αM(t) = (1-\beta t)^{-\alpha}

Var[X]=αβ2Var[X] = \alpha \beta^2

Details​

The expected value of XX can be computed as follows:

E[X]=∫−∞∞xf(x)dx=∫0∞xxα−1e−x/βΓ(α)βαdx=Γ(α+1)βα+1Γ(α)βα∫0∞x(α+1)−1e−x/βΓ(α+1)βα+1dx=αΓ(α)βα+1Γ(α)βα\begin{aligned} E[X] &= \displaystyle\int_{-\infty}^{\infty} xf(x)dx \\ &= \displaystyle\int_{0}^{\infty} x \displaystyle\frac{x^{\alpha -1} e^{-x/\beta}} {\Gamma (\alpha) \beta^{\alpha}} dx \\ &= \displaystyle\frac{\Gamma(\alpha+1)\beta^{\alpha+1}}{\Gamma(\alpha)\beta^{\alpha}} \displaystyle\int_{0}^{\infty} \displaystyle\frac{x^{(\alpha+1) -1} e^{-x/\beta}} {\Gamma (\alpha+1) \beta^{\alpha+1}} dx \\ &= \displaystyle\frac{\alpha\Gamma(\alpha)\beta^{\alpha+1}}{\Gamma(\alpha)\beta^{\alpha}} \end{aligned}

so E[X]=αβE[X] = \alpha\beta

Next, the moment-generating function is given by

E[etX]=∫0∞etxxα−1e−x/βΓ(α)βαdx=1Γ(α)βα∫0∞xα−1etx−x/βdx=Γ(α)ϕαΓ(α)βα∫0∞x(α−1)e−x/ϕΓ(α)ϕαdx\begin{aligned} E[e^{tX}] &= \displaystyle\int_{0}^{\infty} e^{tx} \displaystyle\frac{x^{\alpha-1}e^{-x/\beta}} {\Gamma(\alpha)\beta^{\alpha}} dx \\ &= \displaystyle\frac{1}{\Gamma(\alpha)\beta^{\alpha}} \displaystyle\int_{0}^{\infty} x^{\alpha-1} e^{tx - x/\beta} dx \\ &= \displaystyle\frac{\Gamma(\alpha) \phi^{\alpha} } {\Gamma(\alpha) \beta^{\alpha}} \displaystyle\int_{0}^{\infty} \displaystyle\frac{x^{(\alpha-1)} e^{-x/\phi}} {\Gamma (\alpha) \phi^{\alpha}}dx \end{aligned}

if we choose ϕ\phi so that −xϕ=tx−x/β\displaystyle\frac{-x}{\phi} = tx - x/\beta i.e. −1ϕ=t−1β\displaystyle\frac{-1}{\phi} = t - \displaystyle\frac{1}{\beta} i.e. ϕ=−1t−1/β=β1−βt\phi = - \displaystyle\frac{1}{t-1/\beta} = \displaystyle\frac{\beta}{1 - \beta t} then we have

M(t)=(ϕβ)α=(β/(1−βt)β)α=1(1−βt)α\begin{aligned} M(t) &= \left(\displaystyle\frac{\phi}{\beta}\right)^{\alpha} \\ &= \left(\displaystyle\frac{\beta / (1-\beta t)}{\beta}\right)^{\alpha} \\ &= \displaystyle\frac{1}{(1-\beta t)^{\alpha}} \end{aligned}

or M(t)=(1−βt)−αM(t) = (1-\beta t)^{-\alpha}. +It follows that

M′(t)=(−α)(1−βt)−α−1(−β)=αβ(1−βt)−α−1M'(t) = (-\alpha) (1-\beta t)^{-\alpha-1} (-\beta) = \alpha\beta(1-\beta t)^{-\alpha-1}

so M′(0)=αβM'(0) = \alpha\beta. +Further,

M′′(t)=αβ(−α−1)(1−βt)−α−2(−β)=αβ2(α+1)(1−βt)−α−2\begin{aligned} M''(t) &= \alpha\beta (-\alpha-1)(1-\beta t)^{-\alpha-2} (-\beta) \\ &= \alpha\beta^2 (\alpha+1)(1-\beta t)^{-\alpha-2} \end{aligned}
E[X2]=M′′(0)=αβ2(α+1)=α2β2+αβ2\begin{aligned} E[X^2] &= M''(0) \\ &= \alpha\beta^2 (\alpha+1) \\ &= \alpha^2 \beta^2 + \alpha\beta^2 \end{aligned}

Hence,

Var[X]=E[X]2−E[X]2=α2β2+αβ2−(αβ)2=αβ2\begin{aligned} Var[X] &= E[X]^2 - E[X]^2 \\ &= \alpha^2 \beta^2 + \alpha \beta^2 - (\alpha\beta)^2 \\ &= \alpha \beta^2 \end{aligned}

Special Cases of the Gamma Distribution: The Exponential and Chi-squared Distributions​

Consider the gamma density,

f(x)=xα−1e−xβΓ(α)βα,x>0f(x) = \displaystyle\frac {x^{\alpha - 1} e^{\displaystyle\frac{-x}{\beta}}} {\Gamma(\alpha) \beta^{\alpha}}, x > 0

For parameters α,β>0\alpha, \beta > 0

If α=1\alpha = 1 then

f(x)=1βe−xβ,x>0f(x) = \displaystyle\frac {1} {\beta} e^{\displaystyle\frac{-x}{\beta}}, x > 0

and this is the density of exponential distribution

Consider next the case α=v2\alpha = \displaystyle\frac{v}{2} and β=2\beta = 2 where vv

is an integer, so the density becomes,

f(x)=xv2−1e−x2Γ(v2)Zv2,x>0f(x) = \displaystyle\frac {x^ {\displaystyle\frac{v}{2}- 1} e^{\displaystyle\frac{-x}{2}}} {\Gamma (\displaystyle\frac{v}{2}) Z^{\displaystyle\frac{v}{2}}}, x > 0

This is the density of a chi-squared random variable with vv degrees of freedom.

Details​

Consider, α=v2\alpha = \displaystyle\frac{v}{2} and β=2\beta = 2 where vv is an integer, so the density becomes,

f(x)=xv2−1e−x2Γ(v2)Zv2,x>0f(x) = \displaystyle\frac {x^ {\displaystyle\frac{v}{2}- 1} e^{\displaystyle\frac{-x}{2}}} {\Gamma (\displaystyle\frac{v}{2}) Z^{\displaystyle\frac{v}{2}}}, x > 0

This is the density of a chi-squared random variable with vv degrees of freedom

This is easy to see by starting with Z∼N(0,1)Z \sim N(0,1) and defining W=Z2W = Z^2 so that the c.d.f. is:

H(w)=P[W≤w]=P[Z2≤w]H _{(w)} = P [W \leq w] = P [ Z^2 \leq w]

=P[−w≤Z≤w]= P [ - \sqrt{w}\leq Z \leq \sqrt{w}]

=1−P[∣Z∣>w]= 1 - P [|Z| > \sqrt{w}]

=1−2p[Z<−w]= 1-2p [Z< - \sqrt{w}]

=1−2∫−αwe−t222wdt=1−2ϕ(w)= 1 - 2 \displaystyle\int_{-\alpha}^{\sqrt{w}} \displaystyle\frac{e \displaystyle\frac{-t^2}{2}} {\sqrt{2w}} dt = 1 - 2\phi (\sqrt{w})

The p.d.f. of ww is therefore

h(w)=H′(w)h(w) = H ^\prime(w)

=0−2ϕ′(w)12w12−1= 0 - 2\phi ^\prime (\sqrt{w}) \displaystyle\frac{1} {2} w ^ {\displaystyle\frac{1} {2} -1}

but

ϕ(x)=∫−αxe−t222Πdtϕ′(x)=ddx∫αxe−t222Πdt=e−x222Π\phi (x) = \displaystyle\int_{-\alpha}^{x} \displaystyle\frac{e \displaystyle\frac{-t^2}{2}} {2\Pi} dt \phi ^\prime (x) = \displaystyle\frac {d}{dx}\displaystyle\int_{\alpha}^{x}\displaystyle\frac{e \displaystyle\frac{-t^2}{2}} {2\Pi} dt = \displaystyle\frac{e \displaystyle\frac{-x^2}{2}} {2\Pi}

So

h[w]=−2e−w22Π.12.w12−1h[w] = -2 \displaystyle\frac{e \displaystyle\frac{-w}{2}} {2\Pi} . \displaystyle\frac {1} {2} . w^{\displaystyle\frac {1}{2} -1}

h[w]=w−12−1e−w22Π,w>0h[w] = \displaystyle\frac{w^ {\displaystyle\frac{-1}{2}-1} e \displaystyle\frac{-w}{2}} {2\Pi}, w > 0

We see that we must have h=fh=f with v=1v = 1. +We have also shown Γ(12)212=2π\Gamma (\displaystyle\frac {1}{2}) 2^{\displaystyle\frac{1}{2}} = \sqrt{2\pi}, i.e. Γ(12)=π\Gamma (\displaystyle\frac {1}{2}) = \sqrt{\pi}.

Hence we have shown the χ2\chi^2 distribution on 1 df to be G(α=v2,β=2)G (\alpha = \displaystyle\frac {v}{2}, \beta = 2) when v=1v = 1.

The Sum of Gamma Variables​

In the general case if X1…Xn∼G(α,β)X_1 \ldots X_n \sim G (\alpha, \beta) are independent identically distributed then X1+X2+…Xn∼G(nα,β)X_1 + X_2 + \ldots X_n \sim G (n\alpha, \beta). +In particular, if X1,X2,…,Xv∼χ2X_1, X_2, \ldots, X_v \sim \chi^2 independent identically distributed then Σi=1vXi∼χv2\Sigma_{i=1}^v X_i \sim \chi^2_{v}.

Details​

If XX and YY are independent identically distributed G(α,β)G (\alpha, \beta), then

MX(t)=MY(t)=1(1−βt)αM_X (t) = M_Y (t) = \displaystyle\frac {1} {(1- \beta t)^\alpha}

and

MX+Y(t)=MX(t)MY(t)=1(1−βt)2αM_{X+Y} (t) = M_X (t) M_Y (t) = \displaystyle\frac {1} {(1- \beta t)^{2 \alpha}}

So

X+Y∼G(2α,β)X + Y \sim G (2\alpha, \beta)

In the general case, if X1…Xn∼G(α,β)X_1 \ldots X_n \sim G (\alpha, \beta) are independent identically distributed then X1+X2+…Xn∼G(nα,β)X_1 + X_2 + \ldots X_n \sim G (n\alpha, \beta). +In particular, if X1,X2,…,Xv∼χ2X_1, X_2, \ldots, X_v \sim \chi^2 independent identically distributed, then ∑i=1vXi∼χv2\displaystyle\sum_{i=1}^v X_i \sim \chi^2_{v}.

+ + + + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/index.html b/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/index.html new file mode 100644 index 0000000..a09b5b8 --- /dev/null +++ b/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/index.html @@ -0,0 +1,16 @@ + + + + + +The Multivariate Normal Distribution and Related Topics | Computing and Calculus for Applied Statistics + + + + +
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The Multivariate Normal Distribution and Related Topics

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The Multivariate Normal Distribution and Related Topics

Transformations of Random Variables​

Recall that if XX is a vector of continuous random variables with a joint probability density function and if Y=h(X)Y=h(X) such that hh is a one-to-one function and continuously differentiable with inverse gg so X=g(Y)X= g(Y), then the density of YY is given by

fY(y)=f(g(y))∣J∣f_Y(y)=f(g(y))|J|

Details​

JJ is the Jacobian determinant of gg. +In particular if Y=AXY=AX then

fY(y)=f(A−1y)∣det(A−1)∣f_Y(y)=f(A^{-1}y)|det(A^{-1})|

if AA has an inverse.

The Multivariate Normal Distribution​

Details​

Consider independent identically distributed random variables, Z1,…,Zn∼N(0,1)Z_1, \ldots,Z_n \sim N(0,1),

Z‾=(Z1⋮Zn)\underline{Z} = \left( \begin{array}{ccc} Z_1 \\ \vdots \\ Z_n \end{array} \right)

and let Y‾=AZ‾+μ‾\underline{Y}=A \underline{Z} + \underline{\mu} where AA is an invertible n×nn \times n matrix and μ‾∈Rn\underline{\mu} \in \mathbb{R}^n is a vector, so Z=A−1(Y−μ‾)Z= A^{-1}(Y-\underline{\mu}).

Then the p.d.f. of YY is given by

fY‾(y‾)=fZ‾(A−1(y‾−μ‾))∣det⁡(A−1)∣f_{\underline{Y}}(\underline{y})= f_{\underline{Z}}(A^{-1}(\underline{y}- \underline{\mu})) \vert \det(A^{-1}) \vert

But the joint p.d.f. of Z‾\underline{Z} is the product of the p.d.f.'s of Z1,…,ZnZ_1, \ldots, Z_n, so fZ‾(z‾)=f(z1)⋅f(z2)⋅…⋅f(zn)f_{\underline{Z}}(\underline{z})= f(z_1) \cdot f(z_2) \cdot \ldots \cdot f(z_n) where

f(zi)=12πe−z22f(z_i) = \displaystyle\frac{1}{\sqrt{2 \pi}} e^{-\displaystyle\frac{z^2}{2}}

and hence

fZ‾(z‾)=∏i=1n12πe−z22=(12π)ne−12Σi=1nzi2=1(2π)n2e−12z‾′z‾\begin{align} f_{\underline{Z}}(\underline{z}) &= \prod_{i=1}^n \displaystyle\frac{1}{\sqrt{2 \pi}} e^{\displaystyle\frac{-z^2}{2}} \\ &= \left( \displaystyle\frac{1}{\sqrt{2 \pi}} \right) ^n e^{-\displaystyle\frac{1}{2} \Sigma_{i=1}^n z_i^2} \\ &= \displaystyle\frac{1}{(2 \pi) ^ {\displaystyle\frac{n}{2}}} e^{-\displaystyle\frac{1}{2} \underline{z}'\underline{z}} \end{align}

since

∑i=1nzi2=∥z‾∥2=z‾⋅z‾=z‾′z‾\displaystyle\sum_{i=1}^n z_i^2 = \Vert \underline{z} \Vert ^2 = \underline{z} \cdot \underline{z} = \underline{z}' \underline{z}

The joint p.d.f. of Y‾\underline{Y} is therefore

fY‾(y‾)=fZ‾(A−1(y‾−μ‾))∣det⁡(A−1)∣=1(2π)n2e−12(A−1(y‾−μ‾))′(A−1(y‾−μ‾))1∣det⁡(A)∣\begin{align} f_{\underline{Y}}(\underline{y}) &= f_{\underline{Z}}(A^{-1}(\underline{y} - \underline{\mu})) \vert \det(A^{-1}) \vert \\ &= \displaystyle\frac{1}{(2 \pi)^{\displaystyle\frac{n}{2}}} e^{-\displaystyle\frac{1}{2}(A^{-1}(\underline{y}-\underline{\mu}))'(A^{-1}(\underline{y}-\underline{\mu}))}\displaystyle\frac{1}{\vert \det(A)\vert} \end{align}

We can write det⁡(AA′)=det(A)2\det(AA')=det(A)^2 so ∣det⁡(A)∣=det(AA′)\vert \det(A)\vert = \sqrt{det(AA')} and if we write Σ=AA′\Sigma=AA', then

∣det⁡(A)∣=∣Σ∣12\vert \det(A) \vert = \vert \boldsymbol{\Sigma} \vert ^ {\displaystyle\frac{1}{2}}

Also, note that

(A−1(y‾−μ‾))′(A−1(y‾−μ‾))=(y‾−μ‾)′(A−1)′A−1(y‾−μ‾)=(y‾−μ‾)′Σ−1(y‾−μ‾)(A^{-1}(\underline{y}-\underline{\mu}))'(A^{-1}(\underline{y}-\underline{\mu})) = (\underline{y} - \underline{\mu})'(A^{-1})' A^{-1}(\underline{y} - \underline{\mu}) = (\underline{y} - \underline{\mu})' \boldsymbol{\Sigma}^{-1}(\underline{y}-\underline{\mu})

We can now write

fY‾(y‾)=1(2π)n2∣Σ∣12e−12(y‾−μ‾)Σ−1(y‾−μ‾)f_{\underline{Y}}(\underline{y}) = \displaystyle\frac{1}{(2 \pi)^{\displaystyle\frac{n}{2}} \vert \boldsymbol{\Sigma} \vert ^{\displaystyle\frac{1}{2}}} e^{-\displaystyle\frac{1}{2} (\underline{y}-\underline{\mu}) \boldsymbol{\Sigma}^{-1} (\underline{y}-\underline{\mu})}

This is the density of the multivariate normal distribution.

Note that

E[Y‾]=μE[\underline{Y}] = \mu

Var[Y‾]=Var[AZ‾]=AVar[Z‾]A′=AIA′=ΣVar[\underline{Y}] = Var[A\underline{Z}] = AVar[\underline{Z}]A' = AIA' = \boldsymbol{\Sigma}

Notation: Y‾∼N(μ‾,Σ)\underline{Y}\sim N(\underline{\mu}, \boldsymbol{\Sigma})

Univariate Normal Transforms​

The general univariate normal distribution with density

fY(y)=12πσe−(y−μ)22σ2f_Y(y) = \displaystyle\frac{1}{\sqrt{2\pi}\sigma}e^{-\displaystyle\frac{(y-\mu)^2}{2\sigma^2}}

is a special case of the multivariate version.

Details​

Further, if Z∼N(0,1)Z\sim N(0,1), then clearly X=aZ+μ∼N(μ,σ2)X=aZ+\mu \sim N(\mu,\sigma^2), where σ2=a2\sigma^2=a^2.

Transforms to Lower Dimensions​

If Y∼N(μ,Σ)Y\sim N \left ( \boldsymbol{\mu},\boldsymbol{\Sigma} \right ) is a random vector of length nn and AA is an m×nm\times n matrix of rank m≤nm\leq n, then AY∼N(Aμ,AΣA′)AY \sim N(A\mu,A\Sigma A').

Details​

If Y∼N(μ,Σ)Y\sim N \left ( \boldsymbol{\mu},\boldsymbol{\Sigma} \right ) is a random vector of length nn and AA is an m×nm\times n matrix of rank m≤nm\leq n, then AY∼N(Aμ,AΣA′)AY \sim N(A\mu,A\Sigma A'). +To prove this, set up an (n−m)×n(n-m)\times n matrix, BB, so that the n×nn\times n matrix, CC, formed from combining the rows of AA and BB is of full rank nn. +Then it is easy to derive the density of CYCY which also factors nicely into a product, only one of which contains AYAY, which gives the density for AYAY.

The OLS Estimator​

Suppose Y∼N(Xβ,σ2I)Y \sim N(X \beta,\sigma^2 I). +The ordinary least squares estimator, when the n×pn \times p matrix is of full rank, pp, where p≤np\leq n, is:

β^=(X′X)−1X′Y\hat{\beta} = (X'X)^{-1}X'Y

The random variable which describes the process giving the data and estimate is:

b=(X′X)−1X′Yb = (X'X)^{-1}X'Y

It follows that

β^∼N(β,σ2(X′X)−1)\hat{\beta} \sim N(\beta,\sigma^{2}(X'X)^{-1})

Details​

Suppose Y∼N(Xβ,σ2I)Y \sim N(X \beta,\sigma^2I). +The ordinary least squares estimator, when the n×pn \times p matrix is of full rank, pp, is:

β^=(X′X)−1X′Y\hat{\beta} = (X'X)^{-1}X'Y

The equation below is the random variable which describes the process giving the data and estimate:

b=(X′X)−1X′Yb = (X'X)^{-1}X'Y

If B=(X′X)−1X′B = (X'X)^{-1}X', then we know that

BY∼N(BXβ,B(σ2I)B′)BY \sim N(B X \beta, B(\sigma^{2}I)B')

Note that

BXβ=(X′X)−1X′Xβ=βBX\beta = (X'X)^{-1}X'X\beta=\beta

and

B(σ2I)B′=σ(X′X)−1X′[(X′X)−1X′]′=σ2(X′X)−1X′X(X′X)−1=σ2(X′X)−1\begin{aligned} B(\sigma^{2}I)B' &= \sigma^{}(X'X)^{-1}X'[(X'X)^{-1}X']' \\ &= \sigma^{2}(X'X)^{-1}X'X(X'X)^{-1} \\ &= \sigma^{2}(X'X)^{-1} \end{aligned}

It follows that

β^∼N(β,σ2(X′X)−1)\hat{\beta} \sim N(\beta,\sigma^{2}(X'X)^{-1})

Note

The earlier results regarding the multivariate Gaussian distribution also show that the vector of parameter estimates will be Gaussian even if the original YY-variables are not independent.

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Vector and Matrix Operations

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Vectors and Matrix Operations

Numbers, Vectors, Matrices​

Recall that the set of real numbers is R\mathbb{R} and that a vector, v∈Rnv \in \mathbb{R}^n, is just an nn-tuple of numbers. +Similarly, an n×mn \times m matrix is just a table of numbers, with nn rows and mm columns and we can write

Amn∈RmnA_{mn} \in \mathbb{R}^{mn}

Note that a vector is normally considered equivalent to a n×1n\times 1 matrix i.e. we view these as column vectors.

Examples​

Example

In R, a vector can be generated with:

> X <- 3:6
> X
[1] 3 4 5 6

A matrix can be generated in R as follows,

> matrix(X)
     [,1]
[1,]    3
[2,]    4
[3,]    5
[4,]    6
Note

We note that R distinguishes between vectors and matrices.

Elementary Operations​

We can define multiplication of a real number kk and a vector v=(v1,…,vn)v=(v_1,\ldots,v_n) by k⋅v=(kv1,…,kvn)k\cdot v=(kv_1,\ldots,kv_n). +The sum of two vectors in Rn\mathbb{R}^n, v=(v1,…,vn)v=(v_1,\ldots,v_n) and u=(u1,…,un)u=(u_1,\ldots,u_n) is defined as the vector v+u=(v1+u1,…,vn+un)v+u=(v_1+u_1,\ldots,v_n+u_n). +We can define multiplication of a number and a matrix and the sum of two matrices (of the same sizes) similarly.

Examples​

Example
> A <- matrix(c(1,2,3,4), nr=2, nc=2)

> A
     [,1] [,2]
[1,]    1    3
[2,]    2    4

> B <- matrix(c(1,0,2,1), nr=2, nc=2)

> B
     [,1] [,2]
[1,]    1    2
[2,]    0    1

> A+B
     [,1] [,2]
[1,]    2    5
[2,]    2    5

The Tranpose of a Matrix​

In R, matrices may be constructed using the matrix function and the transpose of AA, A′A^\prime, may be obtained in R by using the t function:

> A <- matrix(1:6, nrow=3)

> t(A)
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6

Details​

If AA is an n×mn \times m matrix with element aija_{ij} in row ii and column jj, then A′A^\prime or ATA^T is the m×nm\times n matrix with element aija_{ij} in row jj and column ii.

Examples​

Example

Consider a vector in R

> x <- 1:4

> x
[1] 1 2 3 4

> t(x)
     [,1] [,2] [,3] [,4]
[1,]    1    2    3    4

> matrix(x)
     [,1]
[1,]    1
[2,]    2
[3,]    3
[4,]    4
> t(matrix(x))
     [,1] [,2] [,3] [,4]
Note

The first solution gives a 1×n1 \times n matrix and the second solution gives a n×1n \times 1 matrix.

Matrix Multiplication​

Matrices AA and BB can be multiplied together if AA is an n×pn \times p matrix and BB is an p×mp\times m matrix. +The general element cijc_{ij} of n×mn\times m, C=ABC=AB, is found by pairing the ithi^{th} row of CC with the jthj^{th} column of BB, and computing the sum of products of the paired terms.

Fig. 39

Details​

Matrices AA and BB can be multiplied together if AA is a n×pn\times p matrix and BB is a p×mp\times m matrix. +Given the general element cijc_{ij} of n×mn \times m matrix, C=ABC=AB is found by pairing the ithi^{th} row of CC with the jthj^{th} column of BB, and computing the sum of products of the paired terms.

Examples​

Example: Matrices in R
> A <- matrix(c(1,3,5,2,4,6),3,2)

> A
     [,1] [,2]
[1,]    1    2
[2,]    3    4
[3,]    5    6

> B <- matrix(c(1,1,2,3),2,2)

> B
     [,1] [,2]
[1,]    1    2
[2,]    1    3

> A%*%B
     [,1] [,2]
[1,]    3    8
[2,]    7   18
[3,]   11   28

More on Matrix Multiplication​

Let AA, BB, and CC be m×nm\times n, n×ln\times l, and l×pl\times p matrices, respectively. +Then we have

(AB)C=A(BC)(AB)C=A(BC)

In general, matrix multiplication is not commutative, that is AB≠BAAB\neq BA.

We also have

(AB)′=B′A′(AB)'=B'A'

In particular, (Av)′(Av)=v′A′Av(Av)'(Av)=v'A'Av, when vv is a n×1n\times1 column vector

More obvious are the rules

  1. A+(B+C)=(A+B)+CA+(B+C)=(A+B)+C

  2. k(A+B)=kA+kBk(A+B)=kA+kB

  3. A(B+C)=AB+ACA(B+C)=AB+AC

where k∈Rk\in\mathbb{R} and when the dimensions of the matrices fit.

Linear Equations​

Details​

General linear equations can be written in the form Ax=bAx=b.

Examples​

Example

The set of equations

2x+3y=42x+3y=4

3x+y=23x+y=2

can be written in matrix formulation as

[2331][xy]=[42]\begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}

i.e. Ax‾=b‾A\underline{x} = \underline{b} for an appropriate choice of A,x‾A, \underline{x} and b‾\underline{b}.

The Unit Matrix​

The n×nn\times n matrix

I=[10…0010⋮⋮0…00…01]I = \left[ \begin{array}{cccc} 1 & 0 & \ldots & 0 \\ 0 & 1 & 0 & \vdots \\ \vdots & 0 & \dots & 0 \\ 0 & \ldots & 0 & 1 \end{array} \right]

is the identity matrix. +This is because if a matrix AA is n×nn\times n

then AI=AA I = A and IA=AI A = A

The Inverse of a Matrix​

If AA is an n×nn \times n matrix and BB is a matrix such that

BA=AB=IBA = AB = I

then BB is said to be the inverse of AA, written

B=A−1B = A ^{-1}

Note that if AA is an n×nn \times n matrix for which an inverse exists, then the equation Ax=bAx = b can be solved and the solution is x=A−1bx = A^{-1} b.

Examples​

Example

If matrix AA is:

[2331]\begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix}

then A−1A ^{-1} is:

[−173737−27]\begin{bmatrix} \displaystyle\frac{-1}{7} & \displaystyle\frac{3}{7} \\ \displaystyle\frac{3}{7} & \displaystyle\frac{-2}{7} \end{bmatrix}
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(max-width:996px){.col{--ifm-col-width:100%;flex-basis:var(--ifm-col-width);margin-left:0}.footer{--ifm-footer-padding-horizontal:0}.colorModeToggle_DEke,.footer__link-separator,.navbar__item,.tableOfContents_bqdL{display:none}.footer__col{margin-bottom:calc(var(--ifm-spacing-vertical)*3)}.footer__link-item{display:block}.hero{padding-left:0;padding-right:0}.navbar>.container,.navbar>.container-fluid{padding:0}.navbar__toggle{display:inherit}.navbar__search-input{width:9rem}.pills--block,.tabs--block{flex-direction:column}.searchBox_ZlJk{position:absolute;right:var(--ifm-navbar-padding-horizontal)}.docItemContainer_F8PC{padding:0 .3rem}}@media (max-width:576px){.markdown h1:first-child{--ifm-h1-font-size:2rem}.markdown>h2{--ifm-h2-font-size:1.5rem}.markdown>h3{--ifm-h3-font-size:1.25rem}}@media (hover:hover){.backToTopButton_sjWU:hover{background-color:var(--ifm-color-emphasis-300)}}@media 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\\displaystyle\\sum_{x=0}^\\infty x p(x) &= \\displaystyle\\sum_{x=0}^\\infty x e^{-\\lambda} \\displaystyle\\frac{\\lambda^x}{x!} \\\\ &= e^{-\\lambda} \\displaystyle\\sum_{x=1}^\\infty \\displaystyle\\frac{x\\lambda^x}{x!} \\\\ &= e^{-\\lambda} \\displaystyle\\sum_{x=1}^\\infty \\displaystyle\\frac{\\lambda^x}{(x-1)!} \\\\ &= e^{-\\lambda} \\lambda \\displaystyle\\sum_{x=1}^\\infty \\displaystyle\\frac{\\lambda^{(x-1)}}{(x-1)!} \\\\ &= e^{-\\lambda} \\lambda \\displaystyle\\sum_{x=0}^\\infty \\displaystyle\\frac{\\lambda^{x}}{x!} \\\\ &= e^{-\\lambda} \\lambda e^{\\lambda} \\\\ &= \\lambda 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samples.")),(0,m.kt)("admonition",{title:"Example",type:"info"},(0,m.kt)("p",{parentName:"admonition"},"How many samples do I need to get a power of ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"0.9")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"0.9")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"0.9"))))),"?"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},'> power.t.test(power=.95, delta=1.5,sd=2, type="one.sample", alternative="one.sided")\n\n One-sample t test power calculation\n\n n = 20.67702\n delta = 1.5\n sd = 2\n sig.level = 0.05\n power = 0.95\n alternative = one.sided\n\nWe would thus need a sample size of $n = 31.15 \\approx 32$ samples to obtain a power of $0.9$ for our analysis.\n'))),(0,m.kt)("admonition",{title:"Example",type:"info"},(0,m.kt)("p",{parentName:"admonition"},"With a sample size of ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"n"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"45")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"n = 45")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"n"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"45"))))),", what will the power of my test be?"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},'> power.t.test(n=45,delta=1.5,sd=2,sig.level=0.05,type="one.sample",alternative="one.sided")\n\n One-sample t test power calculation\n\n n = 45\n delta = 1.5\n sd = 2\n sig.level = 0.05\n power = 0.9995287\n alternative = one.side\n')),(0,m.kt)("p",{parentName:"admonition"},"This is done the same way for two samples only by changing the alternative to ",(0,m.kt)("inlineCode",{parentName:"p"},"two.sample"),".\nFor two sided power analysis, one only needs to change the alternative to ",(0,m.kt)("inlineCode",{parentName:"p"},"two.sided"),".")),(0,m.kt)("admonition",{title:"Example",type:"info"},(0,m.kt)("p",{parentName:"admonition"},"If one is interested in doing a power analysis for an ANOVA test, this is done in a fairly similar way.\nWith a given sample size of ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"n"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"20")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"n=20")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"n"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"20"))))),":"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},"> power.anova.test(groups=4, n=20, between.var=1, within.var=3)\n\n Balanced one-way analysis of variance power calculation\n\n groups = 4\n n = 20\n between.var = 1\n within.var = 3\n sig.level = 0.05\n power = 0.9679022\n")),(0,m.kt)("p",{parentName:"admonition"},"To calculate the sample size needed to obtain a power of ",(0,m.kt)("inlineCode",{parentName:"p"},"0.90")," for a test:"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},"> power.anova.test(groups=4, between.var=1, within.var=3, power=.9)\n\n Balanced one-way analysis of variance power calculation\n\n groups = 4\n n = 15.18834\n between.var = 1\n within.var = 3\n sig.level = 0.05\n power = 0.9\n\nNOTE: n is number in each group\n"))),(0,m.kt)("h2",{id:"the-power-of-the-2-sided-t-test"},"The Power of the 2-sided T-test"),(0,m.kt)("p",null,"A power analysis on a two-sided ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"t")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"t")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6151em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"t"))))),"-test can be done in R using the ",(0,m.kt)("inlineCode",{parentName:"p"},"power.t.test")," command."),(0,m.kt)("h3",{id:"details-6"},"Details"),(0,m.kt)("p",null,"For a one sample test, ",(0,m.kt)("span",{parentName:"p",className:"math 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\\beta_{(\\mu_1\\mu_2)} &= P_{\\mu_1\\mu_2}\\left[\\text{Reject\xa0} H_0 \\right] \\\\ &= P_{\\mu_1\\mu_2}\\left[ \\left|\\displaystyle\\frac{\\overline{X}-\\overline{Y}}{S\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}}\\right| > t^\\ast_{1-\\alpha,n+m-2} \\right] \\\\ &= P_{\\mu_1\\mu_2}\\left[ \\displaystyle\\frac{\\overline{X}-\\overline{Y}}{S\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}} > t^\\ast_{1-\\alpha,n+m-2} \\right] \\\\ &+ P_{\\mu_1\\mu_2}\\left[ \\displaystyle\\frac{\\overline{X}-\\overline{Y}}{S\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}} <-t^\\ast_{1-\\alpha,n+m-2} \\right] \\\\ &= P_{\\mu_1\\mu_2}\\left[ \\displaystyle\\frac{\\displaystyle\\frac{\\overline{X}-\\overline{Y}-(\\mu_1-\\mu_2)}{\\sigma\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}}+\\displaystyle\\frac{(\\mu_1-\\mu_2)}{\\sigma\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}}}{S/\\sqrt{(n+m-2)}} > t^\\ast_{1-\\alpha,n+m-2}\\right] \\\\ &+ P_{\\mu_1\\mu_2}\\left[ \\displaystyle\\frac{\\displaystyle\\frac{\\overline{X}-\\overline{Y}-(\\mu_1-\\mu_2)}{\\sigma\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}}+\\displaystyle\\frac{(\\mu_1-\\mu_2)}{\\sigma\\sqrt{\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{m}}}}{S/\\sqrt{(n+m-2)}} < -t^\\ast_{1-\\alpha,n+m-2}\\right] \\\\ &= P_{\\mu_1\\mu_2}\\left[ \\displaystyle\\frac{Z + \\Delta}{\\sqrt{U/(n+m-2)}} > t^\\ast_{1-\\alpha,n+m-2} \\right] \\\\ &+ P_{\\mu_1\\mu_2}\\left[ \\displaystyle\\frac{Z + \\Delta}{\\sqrt{U/(n+m-2)}} < -t^\\ast_{1-\\alpha,n+m-2} \\right] 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command:"),(0,m.kt)("pre",null,(0,m.kt)("code",{parentName:"pre",className:"language-text"},'source("ancovapow.r")\n')),(0,m.kt)("p",null,"can be used to run the whole thing"),(0,m.kt)("p",null,"The beginning of the command sequence merely consists of comments and definitions of parameter values.\nThese need to be changed for each case separately."),(0,m.kt)("pre",null,(0,m.kt)("code",{parentName:"pre",className:"language-text"},"#\n# ancovapow.r - power computations for analysis of covarariance\n# - one factor, two levels mu0, mu1\n# - one covariate x, x0 stores possible values from which a random set is chosen\n#\n# first set values of parameters\n#\nalpha <- 0.05\nsigma <- 7.5 # the common standard deviation\nx0 <- 20:30 # the set of x values\ndelta <- 10 # the difference in the means\nmu0 <- 0 # the first mean\nmu1 <- mu0+delta # the second mean\nslope <- 2.5 # the slope in the ancova\nJ <- 10 # the common sample size per factor level\nn <- 2*J # the total sample size\nNsim <- 40000 # the number of simulations for power computations\n")),(0,m.kt)("p",null,"Rather than head straight for the ANCOVA, start with a simpler case, namely ignoring the covariate (",(0,m.kt)("inlineCode",{parentName:"p"},"x"),") and merely doing a regular two-sample, two-tailed ",(0,m.kt)("inlineCode",{parentName:"p"},"t"),"-test.\nThis should be reasonably similar to the ANCOVA power computations anyway."),(0,m.kt)("pre",null,(0,m.kt)("code",{parentName:"pre",className:"language-text"},'#\n# Next do the power computations just for a regular two-sided, two-sample t-test\n# and use simulation\n#\nY1 <- matrix(rnorm(J*Nsim,mu0,sigma),ncol=J) # Simulate Nsim samples of size J, ea N(mu1,sigma^2)\nY2 <- matrix(rnorm(J*Nsim,mu1,sigma),ncol=J) # Simulate Nsim samples of size J, ea N(mu2,sigma^2)\ny1mn <- apply(Y1,1,mean) # compute all the simulated y1-means\ny2mn <- apply(Y2,1,mean) # compute all the simulated y2-means\nsy1 <- apply(Y1,1,sd) # compute all the simulated y1-std.devs\nsy2 <- apply(Y2,1,sd) # compute all the simulated y2-std.devs\ns <- sqrt(((J-1)*sy1^2+(J-1)*sy2^2)/(n-2)) # compute all the pooled std.devs\nt <- (y1mn-y2mn)/(s*sqrt(1/J+1/J)) # compute all the Nsim t-statistics\ni <- ifelse(abs(t)>qt(1-alpha/2,n-2),1,0) # for ea t, compute 1=reject, 0=do not reject\npowsim2 <- sum(i)/Nsim # the simulated power\ncat("The simulated power is ",powsim2,"\\n")\n')),(0,m.kt)("p",null,"The above gave the simulated power.\nIn R there is a function to do the same computations and it is worth while to verify the code (and approach) by checking whether these give the same thing:"),(0,m.kt)("pre",null,(0,m.kt)("code",{parentName:"pre",className:"language-text"},'#\n# Then compute the exact power for the t-test\n#\npow2 <- power.t.test(delta=delta,sd=sigma,sig.level=alpha,n=J ,type=c("two.sample"),alternative=c("two.sided"))\ncat("The exact power:\\n")\nprint(pow2)\n')),(0,m.kt)("p",null,"Finally, start setting up the code to do the ANCOVA simulations.\nNote that for this we need to generate the ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," -values.\nIn this example it is assumed that the ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," -values are not under the control of the experimenter but arrive randomly, in the range from 20 to 30 (could e.g. be the age of participants in an experiment)."),(0,m.kt)("pre",null,(0,m.kt)("code",{parentName:"pre",className:"language-text"},"#\n# Finally compute the power in the ANCOVA - note we already have simulated Y1, Y2-values but have not added the x-part yet\n#\nx1 <- matrix(sample(x0,Nsim*J,replace=T),ncol=J) # simulate x-values for y1\nx2 <- matrix(sample(x0,Nsim*J,replace=T),ncol=J) # simulate x-values for y2\nY1 <- Y1+slope*x1\nY2 <- Y2+slope*x2\nfulldat <- cbind(Y1,Y2,x1,x2) # a row now contains all y1, then all y2, then all x1, then all x2; Nsim rows\n")),(0,m.kt)("p",null,'Rather than try to write code to do an ANCOVA, it is natural to use the R function lm to do this.\nThe "trick" below is to extract the ',(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"p")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p")))))," -value from the summary command.\nBy defining a ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mtext",{parentName:"mrow",mathvariant:"monospace"},"wrapper")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\verb|wrapper|")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6528em",verticalAlign:"-0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mord text"},(0,m.kt)("span",{parentName:"span",className:"mord texttt"},"wrapper"))))))," function which takes a single line as an argument, it will subsequently be possible to use the ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mtext",{parentName:"mrow",mathvariant:"monospace"},"apply")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\verb|apply|")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mord text"},(0,m.kt)("span",{parentName:"span",className:"mord texttt"},"apply"))))))," function to extract the ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"p")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p")))))," -values using a one-line R command."),(0,m.kt)("pre",null,(0,m.kt)("code",{parentName:"pre",className:"language-text"},"ancova.pval<-function(onerow){ # extract the ancova p-value for diff in means\n J <- length(onerow)/4\n n <- 2*J\n y <- onerow[1:n] # get the y-data from the row\n x <- onerow[(n+1):(2*n)] # get the x-data from the row\n grps <- factor(c(rep(1,J),rep(2,J))) # define the groups\n sm <- summary(lm(y~x+grps)) # fit the ancova model\n pval <- sm$coefficients[3,4] # extract exactly the right thing from the summary command-the p-value for H0:mu1=mu2\n return(pval)\n}\n")),(0,m.kt)("p",null,"Everything has now been defined so it is possible to compute all the ",(0,m.kt)("span",{parentName:"p",className:"math 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b)\n}\ntruehist(betahat)\n")))}c.isMDXComponent=!0},9863:(a,e,s)=>{s.d(e,{Z:()=>t});const 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math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mi",{parentName:"msup"},"g"),(0,m.kt)("mo",{parentName:"msup",mathvariant:"normal",lspace:"0em",rspace:"0em"},"\u2032")),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mi",{parentName:"msup"},"e"),(0,m.kt)("mrow",{parentName:"msup"},(0,m.kt)("mo",{parentName:"mrow"},"\u2212"),(0,m.kt)("mi",{parentName:"mrow"},"\u03bb"),(0,m.kt)("mi",{parentName:"mrow"},"t")))),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"g'=e^{-\\lambda t}")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.9463em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"g"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8491em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.8491em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 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1")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.9463em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.10764em"}},"f"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.7519em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mtight"},"\u2032"))))))))),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"1")))))," and ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"g"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mo",{parentName:"mrow"},"\u2212"),(0,m.kt)("mstyle",{parentName:"mrow",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mfrac",{parentName:"mstyle"},(0,m.kt)("msup",{parentName:"mfrac"},(0,m.kt)("mi",{parentName:"msup"},"e"),(0,m.kt)("mrow",{parentName:"msup"},(0,m.kt)("mo",{parentName:"mrow"},"\u2212"),(0,m.kt)("mi",{parentName:"mrow"},"\u03bb"),(0,m.kt)("mi",{parentName:"mrow"},"t"))),(0,m.kt)("mi",{parentName:"mfrac"},"\u03bb")))),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"g=-\\displaystyle\\frac{e^{- \\lambda t}}{\\lambda}")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"g"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"2.2121em",verticalAlign:"-0.686em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2212"),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mopen 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mtight"},"t")))))))))))),(0,m.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,m.kt)("span",{parentName:"span"}))))),(0,m.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})))))),".\nWe obtain:"),(0,m.kt)("div",{parentName:"admonition",className:"math math-display"},(0,m.kt)("span",{parentName:"div",className:"katex-display"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mtable",{parentName:"semantics",rowspacing:"0.25em",columnalign:"right 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0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"}))),(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"},(0,m.kt)("mrow",{parentName:"mrow"}),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mstyle",{parentName:"mrow",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mfrac",{parentName:"mstyle"},(0,m.kt)("mn",{parentName:"mfrac"},"1"),(0,m.kt)("mi",{parentName:"mfrac"},"\u03bb")))))))),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\begin{aligned} 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with"),(0,s.kt)("pre",{parentName:"admonition"},(0,s.kt)("code",{parentName:"pre",className:"language-text"},"plotwtle(105)\n"))),(0,s.kt)("h2",{id:"modularity-and-files"},"Modularity and Files"),(0,s.kt)("p",null,"It is advisable to split larger projects into several manageable files."),(0,s.kt)("h3",{id:"details-2"},"Details"),(0,s.kt)("p",null,'Once a project reaches more than five lines of code, it should be stored in one or more separate files.\nIn order to combine these files a single "source" command file can be created'),(0,s.kt)("p",null,"Typically function definitions are stored in separate files, so one may have several separate files like: ",(0,s.kt)("inlineCode",{parentName:"p"},"input.r"),", ",(0,s.kt)("inlineCode",{parentName:"p"},"function.r"),", ",(0,s.kt)("inlineCode",{parentName:"p"},"analysis.r"),", ",(0,s.kt)("inlineCode",{parentName:"p"},"output.r"),".\nWhile developing the analysis, the data would only be read once with 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No function is allowed to write anything into this folder")),(0,s.kt)("li",{parentName:"ul"},(0,s.kt)("p",{parentName:"li"},(0,s.kt)("inlineCode",{parentName:"p"},"/src"),": (abbreviation for ",(0,s.kt)("inlineCode",{parentName:"p"},"source(-code)"),") Here you will store all the functions that you programmed.\nYou can decide to store the executable function here as well or, alternatively, have that one in the root project folder")),(0,s.kt)("li",{parentName:"ul"},(0,s.kt)("p",{parentName:"li"},(0,s.kt)("inlineCode",{parentName:"p"},"/doc"),": Contains further documentation material about your project.\nThis could be, for example, readme files for other people who use your functions, or the paper you wrote about the project, or the ",(0,s.kt)("span",{parentName:"p",className:"math 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",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," is the explanatory variable."),(0,m.kt)("p",null,(0,m.kt)("img",{alt:"Fig. 40",src:s(1606).Z,width:"432",height:"432"})),(0,m.kt)("p",null,"Figure: Example taken from R of a fitted model using linear 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variable."),(0,m.kt)("p",null,"The estimated coefficients describe the change in the dependent variable when there is a single unit increase in the explanatory variable given that everything else is held constant."),(0,m.kt)("p",null,"The standard error is a measure of accuracy and is used to construct the confidence interval.\nConfidence intervals provide a range of values for which there is a set level of confidence that the true population mean will be within the given range.\nFor example, if the CI is set at 95% percent then the probability of observing a value outside the given CI range is less than 0.05"),(0,m.kt)("p",null,"The ",(0,m.kt)("span",{parentName:"p",className:"math 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list(c("a","b"),c("A","B","C"))\n\n> mat\n A B C\na 1 3 5\nb 2 4 6\n\n> mat["b",c("B","C")]\nB C\n4 6\n'))),(0,m.kt)("h2",{id:"the-apply-command"},"The ",(0,m.kt)("inlineCode",{parentName:"h2"},"apply")," Command"),(0,m.kt)("p",null,(0,m.kt)("span",{parentName:"p",className:"math 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",(0,m.kt)("inlineCode",{parentName:"h2"},"tapply")," Command"),(0,m.kt)("p",null,"Commonly one has a data vector and another vector of the same length giving categories for the measurements.\nIn this case one often wants to compute the mean or variance (or median etc) within each category.\nTo do this we use the ",(0,m.kt)("inlineCode",{parentName:"p"},"tapply")," command in R."),(0,m.kt)("h3",{id:"examples-2"},"Examples"),(0,m.kt)("admonition",{title:"Example",type:"info"},(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},'> z <- c(5,7,2,9,3,4,8)\n> i <- c("m","f","m","m","f","m","f")\n')),(0,m.kt)("p",{parentName:"admonition"},"A. Find the sum within each group"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},"> tapply(z,i,sum)\n f m\n18 20\n")),(0,m.kt)("p",{parentName:"admonition"},"B. Find the sample sizes"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},"> tapply(z,i,length)\nf m\n3 4\n")),(0,m.kt)("p",{parentName:"admonition"},"C. Store outputs and use names"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},'> n <- tapply(z,i,length)\n\n> n\nf m\n3 4\n\n> n["m"]\nm\n4\n'))),(0,m.kt)("h2",{id:"logical-indexing"},"Logical Indexing"),(0,m.kt)("p",null,"A logical vector consists of ",(0,m.kt)("span",{parentName:"p",className:"math 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\\displaystyle\\binom{n}{x}p_n^x(1-p_n)^{n-x} &= \\displaystyle\\frac{n!}{x!(n-x!)} \\left ( \\displaystyle\\frac{\\lambda}{n} \\right )^x \\left ( 1-\\displaystyle\\frac{\\lambda}{n} \\right )^{n-x} \\\\ &= \\displaystyle\\frac{N(n-1)(n-2)\\cdots (n-x+1)}{x!} \\displaystyle\\frac{\\displaystyle\\frac{\\lambda}{n}^x}{\\left ( 1-\\displaystyle\\frac{\\lambda}{n} \\right ) ^x} \\left ( 1-\\displaystyle\\frac{\\lambda}{n} \\right )^n \\\\ &= \\displaystyle\\frac{N(n-1)(n-2)\\cdots (n-x+1)}{x!n^x} \\displaystyle\\frac{\\lambda^x}{\\left ( 1-\\displaystyle\\frac{\\lambda}{n} \\right ) ^x} \\left ( 1-\\displaystyle\\frac{\\lambda}{n} \\right )^n 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math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"p")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"))))),", then the joint probability of a given set of numbers is:"),(0,m.kt)("p",null,(0,m.kt)("span",{parentName:"p",className:"math 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\\cdot p(x_2) \\cdot p(x_3) \\cdot \\ldots \\cdot p(x_n)")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"mopen"},"("),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t 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mtight"},"2")))),(0,m.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.kt)("span",{parentName:"span"})))))),(0,m.kt)("span",{parentName:"span",className:"mclose"},")"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mbin"},"\u22c5"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"mopen"},"("),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mtight"},"3")))),(0,m.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.kt)("span",{parentName:"span"})))))),(0,m.kt)("span",{parentName:"span",className:"mclose"},")"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mbin"},"\u22c5"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4445em"}}),(0,m.kt)("span",{parentName:"span",className:"minner"},"\u2026"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mbin"},"\u22c5"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"mopen"},"("),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.1514em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal mtight"},"n")))),(0,m.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.kt)("span",{parentName:"span"})))))),(0,m.kt)("span",{parentName:"span",className:"mclose"},")")))))),(0,m.kt)("p",null,"Suppose each probability includes the same parameter ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"\u03b8")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\theta")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"\u03b8"))))),", then this is typically written:"),(0,m.kt)("p",null,(0,m.kt)("span",{parentName:"p",className:"math 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\\ldots, {p_{\\theta}}(x_n)")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 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mtight",style:{marginRight:"0.02778em"}},"\u03b8"))))),(0,m.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.kt)("span",{parentName:"span"}))))))),(0,m.kt)("span",{parentName:"span",className:"mopen"},"("),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.1514em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.kt)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal mtight"},"n")))),(0,m.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.kt)("span",{parentName:"span"})))))),(0,m.kt)("span",{parentName:"span",className:"mclose"},")")))))),(0,m.kt)("p",null,"Now consider the set of outcomes ",(0,m.kt)("span",{parentName:"p",className:"math 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f(p) &= \\ln(p^{x} (1-p)^{n-x}) \\\\ &= \\ln p^{x} + \\ln(1-p)^{n-x} \\\\ &= x \\ln p + (n-x) \\ln (1-p) \\end{aligned}")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"4.5em",verticalAlign:"-2em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mtable"},(0,m.kt)("span",{parentName:"span",className:"col-align-r"},(0,m.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.kt)("span",{parentName:"span",className:"vlist-r"},(0,m.kt)("span",{parentName:"span",className:"vlist",style:{height:"2.5em"}},(0,m.kt)("span",{parentName:"span",style:{top:"-4.66em"}},(0,m.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord 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in the ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"x"),(0,n.kt)("mi",{parentName:"mrow"},"y")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"xy")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"y"))))),"-plane using R:"),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},'plot(2,4,xlim=c(0,6),ylim=c(0,6),xlab="x",ylab="y",cex=2)\ntext(2,4,"(2,4)",pos=4,cex=2)\n')),(0,n.kt)("p",{parentName:"admonition"},"Additional points can be added using the ",(0,n.kt)("inlineCode",{parentName:"p"},"points()")," function:"),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},"points(3,5, cex = 0.5) ## a point at (3,5)\n"))),(0,n.kt)("p",null,"If you have two sets of coordinates on a plane you, can calculate the distance between the two points and graph the line connecting the points."),(0,n.kt)("admonition",{title:"Example",type:"info"},(0,n.kt)("p",{parentName:"admonition"},"What is the distance between the two points ",(0,n.kt)("span",{parentName:"p",className:"math 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reset-size6 size3 mtight"},(0,n.kt)("span",{parentName:"span",className:"mord mtight"},(0,n.kt)("span",{parentName:"span",className:"mord 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",(0,m.kt)("inlineCode",{parentName:"p"},"T"),", ",(0,m.kt)("inlineCode",{parentName:"p"},"H"),", ..., ",(0,m.kt)("inlineCode",{parentName:"p"},"T"),"."),(0,m.kt)("p",null,"How many of these possible sequences contain exactly ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," tails?\nThere are ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"n")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"n")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"n")))))," positions in the sequence, we can choose ",(0,m.kt)("span",{parentName:"p",className:"math 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math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"y"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mi",{parentName:"mrow"},"a"),(0,m.kt)("mo",{parentName:"mrow"},"+"),(0,m.kt)("mi",{parentName:"mrow"},"b"),(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"y = a + bx")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord 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mathnormal"},"x"))))),", then we can find the value of ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," by considering ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"y"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mi",{parentName:"mrow"},"a"),(0,m.kt)("mo",{parentName:"mrow"},"+"),(0,m.kt)("mi",{parentName:"mrow"},"b"),(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"y = a+bx")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"y"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"a"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"b"),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," as an equation to be solved for ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"))))),", since ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"y")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"y")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"y"))))),", ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"a")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"a")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"a")))))," and ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"b")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"b")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"b")))))," are all known."),(0,m.kt)("p",null,"The general solution is found through the following steps:"),(0,m.kt)("ol",null,(0,m.kt)("li",{parentName:"ol"},(0,m.kt)("p",{parentName:"li"},"Equation: ",(0,m.kt)("span",{parentName:"p",className:"math 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mathnormal"},"x"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7778em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"y"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,m.kt)("span",{parentName:"span",className:"mbin"},"\u2212"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord 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x &= \\displaystyle\\frac{-(-3) \\pm \\sqrt{(-3)^2 - 4 \\cdot 1 \\cdot 2}}{2 \\cdot 1} \\\\ x &= \\displaystyle\\frac{3 \\pm \\sqrt{9 - 8}}{2} \\\\ x &= \\displaystyle\\frac{3 \\pm \\sqrt{1}}{2} \\\\ x &= \\displaystyle\\frac{3 + 1}{2} \\text{ or\xa0} \\displaystyle\\frac{3 - 1}{2} \\\\ x &= \\displaystyle\\frac{4}{2} \\text{ or\xa0} \\displaystyle\\frac{2}{2} \\\\ x &= 2 \\text{ or\xa0} 1 \\end{aligned}")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"13.8297em",verticalAlign:"-6.6648em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mtable"},(0,m.kt)("span",{parentName:"span",className:"col-align-r"},(0,m.kt)("span",{parentName:"span",className:"vlist-t 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With R"),(0,n.kt)("p",null,"Download R from the R website: ",(0,n.kt)("a",{parentName:"p",href:"http://www.r-project.org/"},"http://www.r-project.org/")),(0,n.kt)("p",null,"Look at on-line information on R, and take the tutor-web R tutorial: ",(0,n.kt)("a",{parentName:"p",href:"http://tutor-web.net/stats/stats240.1"},"http://tutor-web.net/stats/stats240.1")),(0,n.kt)("p",null,"Simple R commands:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Assignment:"),(0,n.kt)("pre",{parentName:"li"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},"x <- 2\n"))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Arithmetic:"),(0,n.kt)("pre",{parentName:"li"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},"2*5+4\n")))),(0,n.kt)("h3",{id:"details-1"},"Details"),(0,n.kt)("p",null,"To assign values to a variable in R one can use ",(0,n.kt)("inlineCode",{parentName:"p"},"<-")," or ",(0,n.kt)("inlineCode",{parentName:"p"},"="),".\nhowever, these are ",(0,n.kt)("strong",{parentName:"p"},"NOT")," equivalent.\nUsing the equals sign is confusing and therefore not recommended."),(0,n.kt)("h3",{id:"examples-1"},"Examples"),(0,n.kt)("admonition",{title:"Example",type:"info"},(0,n.kt)("p",{parentName:"admonition"},"Assigning values to a variable:"),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},"x <- 2\ny <- 3\nz <- x+y\n"))),(0,n.kt)("admonition",{title:"Example: Viewing assigned values",type:"info"},(0,n.kt)("p",{parentName:"admonition"},"Type the name, i.e. ",(0,n.kt)("inlineCode",{parentName:"p"},"z"),", to view the assigned value."),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},"> z\n[1] 5\n"))),(0,n.kt)("h2",{id:"the-integers"},"The Integers"),(0,n.kt)("p",null,"The set of positive and negative 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compute"),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mn",{parentName:"mrow"},"15"),(0,n.kt)("mo",{parentName:"mrow"},"\u2212"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,n.kt)("mn",{parentName:"mrow"},"24"),(0,n.kt)("mo",{parentName:"mrow"},"+"),(0,n.kt)("mn",{parentName:"mrow"},"36"),(0,n.kt)("mo",{parentName:"mrow",stretchy:"false"},")")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"15 - (24 + 36)")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"15"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"\u2212"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mopen"},"("),(0,n.kt)("span",{parentName:"span",className:"mord"},"24"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"+"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"36"),(0,n.kt)("span",{parentName:"span",className:"mclose"},")")))))),(0,n.kt)("p",{parentName:"admonition"},"we first note that the parentheses (brackets) imply a precedence;\nanything inside brackets should be evaluated first.\nThus, we first add ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mn",{parentName:"mrow"},"36")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"36")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"36")))))," to ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mn",{parentName:"mrow"},"24")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"24")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"24")))))," and then we subtract that from ",(0,n.kt)("span",{parentName:"p",className:"math 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that the answer is a negative number.")),(0,n.kt)("admonition",{title:"Example",type:"info"},(0,n.kt)("p",{parentName:"admonition"},"Simple arithmetic in R is easily done at the command prompt:"),(0,n.kt)("pre",{parentName:"admonition"},(0,n.kt)("code",{parentName:"pre",className:"language-text"},"> 79-8*3\n[1] 55\n\n> 15-(24+36)\n[1] -45\n"))),(0,n.kt)("h2",{id:"rational-numbers"},"Rational Numbers"),(0,n.kt)("p",null,(0,n.kt)("strong",{parentName:"p"},"Rational numbers")," are fractions denoted ",(0,n.kt)("span",{parentName:"p",className:"math 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math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"q")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"q")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"q")))))," are integers.\nWe can simplify fractions if the numerator and denominator contain common terms."),(0,n.kt)("h3",{id:"details-3"},"Details"),(0,n.kt)("p",null,(0,n.kt)("img",{alt:"Fig. 1",src:s(1633).Z,width:"432",height:"432"})),(0,n.kt)("admonition",{title:"Definition",type:"note"},(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("strong",{parentName:"p"},"Rational numbers")," are fractions denoted ",(0,n.kt)("span",{parentName:"p",className:"math 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\\cdot 2}{3 \\cdot 2} + \\displaystyle\\frac{1 \\cdot 3}{2 \\cdot 3} = \\displaystyle\\frac{2}{6} + \\displaystyle\\frac{3}{6} = \\displaystyle\\frac{5}{6}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t 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vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"5"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"6")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"75"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"96")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.363em",verticalAlign:"-0.936em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.427em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"75/3"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"96/3")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.936em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"25"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"32")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"}))))))),(0,n.kt)("p",{parentName:"admonition"},"We can see that in the last step of the equation, the factor has been simplified.\nTo do this we use factoring.\nHere we obtain:"),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mstyle",{parentName:"mrow",scriptlevel:"0",displaystyle:"true"},(0,n.kt)("mfrac",{parentName:"mstyle"},(0,n.kt)("mn",{parentName:"mfrac"},"96"),(0,n.kt)("mn",{parentName:"mfrac"},"75")),(0,n.kt)("mo",{parentName:"mstyle"},"="),(0,n.kt)("mstyle",{parentName:"mstyle",scriptlevel:"0",displaystyle:"true"},(0,n.kt)("mfrac",{parentName:"mstyle"},(0,n.kt)("mrow",{parentName:"mfrac"},(0,n.kt)("mn",{parentName:"mrow"},"3"),(0,n.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,n.kt)("mn",{parentName:"mrow"},"32")),(0,n.kt)("mrow",{parentName:"mfrac"},(0,n.kt)("mn",{parentName:"mrow"},"3"),(0,n.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,n.kt)("mn",{parentName:"mrow"},"25")))))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\displaystyle\\frac{96}{75} = \\displaystyle\\frac{3 \\cdot 32}{3 \\cdot 25}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"75"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"96")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"3"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"\u22c5"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"25"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"3"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mbin"},"\u22c5"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"32")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"}))))))),(0,n.kt)("p",{parentName:"admonition"},"We can now remove ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mn",{parentName:"mrow"},"3")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"3")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"3"))))),", or the multiplier, as it is on both sides of the fraction.\nSo we have:"),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mstyle",{parentName:"mrow",scriptlevel:"0",displaystyle:"true"},(0,n.kt)("mfrac",{parentName:"mstyle"},(0,n.kt)("mn",{parentName:"mfrac"},"32"),(0,n.kt)("mn",{parentName:"mfrac"},"25")),(0,n.kt)("mo",{parentName:"mstyle"},"="),(0,n.kt)("mstyle",{parentName:"mstyle",scriptlevel:"0",displaystyle:"true"},(0,n.kt)("mfrac",{parentName:"mstyle"},(0,n.kt)("mn",{parentName:"mfrac"},"25"),(0,n.kt)("mn",{parentName:"mfrac"},"25")),(0,n.kt)("mo",{parentName:"mstyle"},"+"),(0,n.kt)("mstyle",{parentName:"mstyle",scriptlevel:"0",displaystyle:"true"},(0,n.kt)("mfrac",{parentName:"mstyle"},(0,n.kt)("mn",{parentName:"mfrac"},"7"),(0,n.kt)("mn",{parentName:"mfrac"},"25")),(0,n.kt)("mo",{parentName:"mstyle"},"="),(0,n.kt)("mstyle",{parentName:"mstyle",scriptlevel:"0",displaystyle:"true"},(0,n.kt)("mfrac",{parentName:"mstyle"},(0,n.kt)("mn",{parentName:"mfrac"},"32"),(0,n.kt)("mn",{parentName:"mfrac"},"25"))))))),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\displaystyle\\frac{32}{25} = \\displaystyle\\frac{25}{25} + \\displaystyle\\frac{7}{25} = \\displaystyle\\frac{32}{25}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"25"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"32")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose 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vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"1"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"2")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))),(0,n.kt)("p",{parentName:"admonition"},"These functions always provide the same result and therefore we can turn the fraction we are dividing by upside down and multiply it to the other fraction as that is usually much easier."),(0,n.kt)("p",{parentName:"admonition"},"We can therefore rewrite"),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math 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\\displaystyle\\frac{1}{5}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t 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\\div \\displaystyle\\frac{10}{6} = \\displaystyle\\frac{29}{12} \\cdot \\displaystyle\\frac{6}{10} = \\displaystyle\\frac{174}{120}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mopen nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t 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nulldelimiter"}))))))),(0,n.kt)("p",{parentName:"admonition"},"and then dividing those numbers (",(0,n.kt)("inlineCode",{parentName:"p"},"87")," and ",(0,n.kt)("inlineCode",{parentName:"p"},"60"),") by ",(0,n.kt)("inlineCode",{parentName:"p"},"3"),", since they can't be divided by ",(0,n.kt)("inlineCode",{parentName:"p"},"2"),".\nDividing by ",(0,n.kt)("inlineCode",{parentName:"p"},"3")," gives you"),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math 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nulldelimiter"}),(0,n.kt)("span",{parentName:"span",className:"mfrac"},(0,n.kt)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"120"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"174")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose 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vlist-t2"},(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"1.3214em"}},(0,n.kt)("span",{parentName:"span",style:{top:"-2.314em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"3"))),(0,n.kt)("span",{parentName:"span",style:{top:"-3.23em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,n.kt)("span",{parentName:"span",style:{top:"-3.677em"}},(0,n.kt)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},(0,n.kt)("span",{parentName:"span",className:"mord"},"3")))),(0,n.kt)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,n.kt)("span",{parentName:"span",className:"vlist-r"},(0,n.kt)("span",{parentName:"span",className:"vlist",style:{height:"0.686em"}},(0,n.kt)("span",{parentName:"span"}))))),(0,n.kt)("span",{parentName:"span",className:"mclose nulldelimiter"})),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,n.kt)("span",{parentName:"span",className:"mord"},"1")))))),(0,n.kt)("p",{parentName:"admonition"},"The final answer therefore is"),(0,n.kt)("p",{parentName:"admonition"},(0,n.kt)("span",{parentName:"p",className:"math 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on the number line.\nFilling in the missing points to obtain a continuum results in the set of ",(0,n.kt)("strong",{parentName:"p"},"real numbers"),"."),(0,n.kt)("p",null,"Denoted by ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow",mathvariant:"double-struck"},"R")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\mathbb{R}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6889em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathbb"},"R")))))," the entire set of real numbers 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mathnormal",style:{marginRight:"0.07153em"}},"C")))))," is the circumference of a circle and ",(0,n.kt)("span",{parentName:"p",className:"math math-inline"},(0,n.kt)("span",{parentName:"span",className:"katex"},(0,n.kt)("span",{parentName:"span",className:"katex-mathml"},(0,n.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,n.kt)("semantics",{parentName:"math"},(0,n.kt)("mrow",{parentName:"semantics"},(0,n.kt)("mi",{parentName:"mrow"},"D")),(0,n.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"D")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"D")))))," is the diameter and we define ",(0,n.kt)("span",{parentName:"p",className:"math 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= \\sum_{n=0}^{\\infty }\\displaystyle\\frac{1}{n!}")))),(0,n.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,n.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.kt)("span",{parentName:"span",className:"mrel"},"="),(0,n.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,n.kt)("span",{parentName:"span",className:"base"},(0,n.kt)("span",{parentName:"span",className:"strut",style:{height:"2.0074em",verticalAlign:"-0.686em"}}),(0,n.kt)("span",{parentName:"span",className:"mop"},(0,n.kt)("span",{parentName:"span",className:"mop op-symbol 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with unit side length, what is the length of its hypotenuse and what class of numbers does it belong to?"),(0,n.kt)("p",{parentName:"admonition"},"An isosceles triangle is defined as having adjacent and opposite sides of same length, connected by a ",(0,n.kt)("span",{parentName:"p",className:"math 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can be calculated in R using the code below:"),(0,m.kt)("pre",{parentName:"admonition"},(0,m.kt)("code",{parentName:"pre",className:"language-text"},"> dbinom(3,30,0.4)\n[1] 0.0002659437\n")),(0,m.kt)("p",{parentName:"admonition"},"This is equal to the manual calculation using the binomial theorem.")),(0,m.kt)("admonition",{title:"Example",type:"info"},(0,m.kt)("p",{parentName:"admonition"},"Suppose a youngster puts his shirt on by himself every day for five days.\nThe probability that he puts it on the right way each time is ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"p"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"0.2")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p=0.2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"0.2"))))),".\nWe let ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"X")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"X")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.07847em"}},"X")))))," be a random variable that describes the number of times the youngster puts his shirt on the right way.\nThe youngster can either put the shirt on the wrong or the right way so ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"X")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"X")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.07847em"}},"X")))))," follows the binomial distribution with the parameters ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"p"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"0.2")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p=0.2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"0.2")))))," (the probability of a successful trial) and ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"n"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"5")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"n=5")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"n"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"5")))))," (number of trials).\nWe can now calculate for example the probability that the youngster will put it on the right way for at least four days"),(0,m.kt)("p",{parentName:"admonition"},"Putting the shirt on the right way for at least four days, means that the youngster will either put it on the right way for either four or five days (at least four or more days of five days total).\nWe thus have to calculate the probability that the youngster will put his shirt on the right way for four and five days separately and then we add it together.\nWe can write this process as follows:"),(0,m.kt)("div",{parentName:"admonition",className:"math math-display"},(0,m.kt)("span",{parentName:"div",className:"katex-display"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mtable",{parentName:"semantics",rowspacing:"0.25em",columnalign:"right 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(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"8"),(0,m.kt)("mn",{parentName:"msup"},"0")))))),(0,m.kt)("mtr",{parentName:"mtable"},(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"}))),(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"},(0,m.kt)("mrow",{parentName:"mrow"}),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"5"),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"2"),(0,m.kt)("mn",{parentName:"msup"},"4")),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"0.8"),(0,m.kt)("mo",{parentName:"mrow"},"+"),(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"2"),(0,m.kt)("mn",{parentName:"msup"},"5")),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"1"))))),(0,m.kt)("mtr",{parentName:"mtable"},(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"}))),(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"},(0,m.kt)("mrow",{parentName:"mrow"}),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"5"),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"0.8"),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"2"),(0,m.kt)("mn",{parentName:"msup"},"4")),(0,m.kt)("mo",{parentName:"mrow"},"+"),(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"2"),(0,m.kt)("mn",{parentName:"msup"},"5")))))),(0,m.kt)("mtr",{parentName:"mtable"},(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"}))),(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"},(0,m.kt)("mrow",{parentName:"mrow"}),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"4"),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"2"),(0,m.kt)("mn",{parentName:"msup"},"4")),(0,m.kt)("mo",{parentName:"mrow"},"+"),(0,m.kt)("mn",{parentName:"mrow"},"0."),(0,m.kt)("msup",{parentName:"mrow"},(0,m.kt)("mn",{parentName:"msup"},"2"),(0,m.kt)("mn",{parentName:"msup"},"5")))))),(0,m.kt)("mtr",{parentName:"mtable"},(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"}))),(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"},(0,m.kt)("mrow",{parentName:"mrow"}),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"4"),(0,m.kt)("mo",{parentName:"mrow"},"\u22c5"),(0,m.kt)("mn",{parentName:"mrow"},"0.0016"),(0,m.kt)("mo",{parentName:"mrow"},"+"),(0,m.kt)("mn",{parentName:"mrow"},"0.00032"))))),(0,m.kt)("mtr",{parentName:"mtable"},(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"}))),(0,m.kt)("mtd",{parentName:"mtr"},(0,m.kt)("mstyle",{parentName:"mtd",scriptlevel:"0",displaystyle:"true"},(0,m.kt)("mrow",{parentName:"mstyle"},(0,m.kt)("mrow",{parentName:"mrow"}),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"0.00672")))))),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\begin{aligned} P(X\\geq4) &= P(X=4) + P(X=5) \\\\ &= \\displaystyle\\binom{5}{4}\\cdot0.2^4\\cdot(1-0.2)^{5-4} + \\displaystyle\\binom{5}{5}\\cdot0.2^5\\cdot(1-0.2)^{5-5} \\\\ &= 5\\cdot0.2^4\\cdot0.8^1 + 1\\cdot0.2^5\\cdot0.8^0 \\\\ &= 5\\cdot0.2^4\\cdot0.8 + 0.2^5\\cdot1 \\\\ & =5\\cdot0.8 \\cdot0.2^4 + 0.2^5 \\\\ &= 4\\cdot0.2^4 + 0.2^5 \\\\ &= 4\\cdot0.0016 + 0.00032 \\\\ &= 0.00672 \\end{aligned}")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"13.2965em",verticalAlign:"-6.3982em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mtable"},(0,m.kt)("span",{parentName:"span",className:"col-align-r"},(0,m.kt)("span",{parentName:"span",className:"vlist-t 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random variable ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"X")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"X")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.07847em"}},"X")))))," has this distribution, then we can write:"),(0,m.kt)("p",null,(0,m.kt)("span",{parentName:"p",className:"math 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p(x)")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.13889em"}},"P"),(0,m.kt)("span",{parentName:"span",className:"mopen"},"["),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.07847em"}},"X"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord 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math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"p"),(0,m.kt)("mo",{parentName:"mrow",stretchy:"false"},"("),(0,m.kt)("mi",{parentName:"mrow"},"x"),(0,m.kt)("mo",{parentName:"mrow",stretchy:"false"},")"),(0,m.kt)("mo",{parentName:"mrow"},"="),(0,m.kt)("mn",{parentName:"mrow"},"0")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p(x) = 0")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"p"),(0,m.kt)("span",{parentName:"span",className:"mopen"},"("),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x"),(0,m.kt)("span",{parentName:"span",className:"mclose"},")"),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.kt)("span",{parentName:"span",className:"mrel"},"="),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"0")))))," if ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"x")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"x")))))," is not one of these ",(0,m.kt)("span",{parentName:"p",className:"math 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have your grade dropped in the following scenarios?"),(0,m.kt)("ol",{parentName:"admonition"},(0,m.kt)("li",{parentName:"ol"},"You set your alarm for 7:15 throughout the duration of the class."),(0,m.kt)("li",{parentName:"ol"},"You set your alarm for 7:15 until you reach 5 missed classes, then switch to 7:10."),(0,m.kt)("li",{parentName:"ol"},"You set your alarm for 7:15 until you reach 5 missed classes, then switch to 7:00.")),(0,m.kt)("p",{parentName:"admonition"},"Part B: What is your expected grade in the course, assuming you would have had a 7/10 without the late penalty, and:"),(0,m.kt)("ol",{parentName:"admonition"},(0,m.kt)("li",{parentName:"ol"},"You would never choose the first alarm-clock strategy and you would most likely choose scenario 2 (let's say 9/10 times), but there's a small chance you might choose the 3rd strategy (let's say 1/10 times)."),(0,m.kt)("li",{parentName:"ol"},"You would never choose the first alarm-clock strategy and you would most likely choose 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