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Applications of Differentiation

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- + \ No newline at end of file diff --git a/Functions/Applications of Differentiation/read/index.html b/Functions/Applications of Differentiation/read/index.html index 91b9035..0be2c61 100644 --- a/Functions/Applications of Differentiation/read/index.html +++ b/Functions/Applications of Differentiation/read/index.html @@ -4,7 +4,7 @@ Applications of Differentiation | Computing and Calculus for Applied Statistics - + @@ -22,7 +22,7 @@ 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}
- + \ No newline at end of file diff --git a/Functions/Continuity and Limits/index.html b/Functions/Continuity and Limits/index.html index b1e27c4..c403ff1 100644 --- a/Functions/Continuity and Limits/index.html +++ b/Functions/Continuity and Limits/index.html @@ -4,13 +4,13 @@ Continuity and Limits | Computing and Calculus for Applied Statistics - +
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Continuity and Limits

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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.

- + \ No newline at end of file diff --git a/Functions/Derivatives/index.html b/Functions/Derivatives/index.html index 3e0f1f1..6e8a369 100644 --- a/Functions/Derivatives/index.html +++ b/Functions/Derivatives/index.html @@ -4,13 +4,13 @@ Derivatives | Computing and Calculus for Applied Statistics - +
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Derivatives

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- + \ No newline at end of file diff --git a/Functions/Derivatives/read/index.html b/Functions/Derivatives/read/index.html index 079fe84..f4e3e4b 100644 --- a/Functions/Derivatives/read/index.html +++ b/Functions/Derivatives/read/index.html @@ -4,7 +4,7 @@ Derivatives | Computing and Calculus for Applied Statistics - + @@ -14,7 +14,7 @@ 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}
- + \ No newline at end of file diff --git a/Functions/Functions of Functions and the Exponential Function/index.html b/Functions/Functions of Functions and the Exponential Function/index.html index 94bf70f..cfe701f 100644 --- a/Functions/Functions of Functions and the Exponential Function/index.html +++ b/Functions/Functions of Functions and the Exponential Function/index.html @@ -4,13 +4,13 @@ Functions of Functions and the Exponential Function | Computing and Calculus for Applied Statistics - +
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Functions of Functions and the Exponential Function

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- + \ No newline at end of file diff --git a/Functions/Functions of Functions and the Exponential Function/read/index.html b/Functions/Functions of Functions and the Exponential Function/read/index.html index 411a031..9e56323 100644 --- a/Functions/Functions of Functions and the Exponential Function/read/index.html +++ b/Functions/Functions of Functions and the Exponential Function/read/index.html @@ -4,7 +4,7 @@ Functions of Functions and the Exponential Function | Computing and Calculus for Applied Statistics - + @@ -15,7 +15,7 @@ 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.

- + \ No newline at end of file diff --git a/Functions/Integrals and Probability Density Functions/index.html b/Functions/Integrals and Probability Density Functions/index.html index 19dc237..b818fcf 100644 --- a/Functions/Integrals and Probability Density Functions/index.html +++ b/Functions/Integrals and Probability Density Functions/index.html @@ -4,13 +4,13 @@ Integrals and Probability Density Functions | Computing and Calculus for Applied Statistics - +
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Integrals and Probability Density Functions

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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

- + \ No newline at end of file diff --git a/Functions/Inverse Functions and the Logarithm/index.html b/Functions/Inverse Functions and the Logarithm/index.html index 276f03c..411279d 100644 --- a/Functions/Inverse Functions and the Logarithm/index.html +++ b/Functions/Inverse Functions and the Logarithm/index.html @@ -4,13 +4,13 @@ Inverse Functions and the Logarithm | Computing and Calculus for Applied Statistics - +
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Inverse Functions and the Logarithm

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- + \ No newline at end of file diff --git a/Functions/Inverse Functions and the Logarithm/read/index.html b/Functions/Inverse Functions and the Logarithm/read/index.html index aef0244..9f75717 100644 --- a/Functions/Inverse Functions and the Logarithm/read/index.html +++ b/Functions/Inverse Functions and the Logarithm/read/index.html @@ -4,7 +4,7 @@ Inverse Functions and the Logarithm | Computing and Calculus for Applied Statistics - + @@ -36,7 +36,7 @@ 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

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- + \ No newline at end of file diff --git a/Functions/Miscellanea/read/index.html b/Functions/Miscellanea/read/index.html index e1adf23..ce36782 100644 --- a/Functions/Miscellanea/read/index.html +++ b/Functions/Miscellanea/read/index.html @@ -4,7 +4,7 @@ Miscellanea | Computing and Calculus for Applied Statistics - + @@ -59,7 +59,7 @@ c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​X−μ0​​

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.

- + \ No newline at end of file diff --git a/Functions/Overview/index.html b/Functions/Overview/index.html index f4450d1..5f9a69e 100644 --- a/Functions/Overview/index.html +++ b/Functions/Overview/index.html @@ -4,13 +4,13 @@ Overview | Computing and Calculus for Applied Statistics - +
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- + \ No newline at end of file diff --git a/Functions/Principles of Programming/index.html b/Functions/Principles of Programming/index.html index 14d3d86..fee5205 100644 --- a/Functions/Principles of Programming/index.html +++ b/Functions/Principles of Programming/index.html @@ -4,13 +4,13 @@ Principles of Programming | Computing and Calculus for Applied Statistics - +
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Principles of Programming

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- + \ No newline at end of file diff --git a/Functions/Principles of Programming/read/index.html b/Functions/Principles of Programming/read/index.html index 5059917..50fdd23 100644 --- a/Functions/Principles of Programming/read/index.html +++ b/Functions/Principles of Programming/read/index.html @@ -4,7 +4,7 @@ Principles of Programming | Computing and Calculus for Applied Statistics - + @@ -23,7 +23,7 @@ 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##### ######################
- + \ No newline at end of file diff --git a/Functions/Sequences and Series/index.html b/Functions/Sequences and Series/index.html index 922dbfa..659f123 100644 --- a/Functions/Sequences and Series/index.html +++ b/Functions/Sequences and Series/index.html @@ -4,13 +4,13 @@ Sequences and Series | Computing and Calculus for Applied Statistics - +
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Sequences and Series

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- + \ No newline at end of file diff --git a/Functions/Sequences and Series/read/index.html b/Functions/Sequences and Series/read/index.html index a9f992e..9122611 100644 --- a/Functions/Sequences and Series/read/index.html +++ b/Functions/Sequences and Series/read/index.html @@ -4,7 +4,7 @@ Sequences and Series | Computing and Calculus for Applied Statistics - + @@ -13,7 +13,7 @@ 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 b0edec7..f4a3968 100644 --- a/Functions/Slopes of Lines and Curves/index.html +++ b/Functions/Slopes of Lines and Curves/index.html @@ -4,13 +4,13 @@ Slopes of Lines and Curves | Computing and Calculus for Applied Statistics - +
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Slopes of Lines and Curves

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- + \ No newline at end of file diff --git a/Functions/Slopes of Lines and Curves/read/index.html b/Functions/Slopes of Lines and Curves/read/index.html index 4e52e70..4cc0b77 100644 --- a/Functions/Slopes of Lines and Curves/read/index.html +++ b/Functions/Slopes of Lines and Curves/read/index.html @@ -4,7 +4,7 @@ Slopes of Lines and Curves | Computing and Calculus for Applied Statistics - + @@ -21,7 +21,7 @@ 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 fb96f53..156de20 100644 --- a/Functions/The Central Limit Theorem and Related Topics/index.html +++ b/Functions/The Central Limit Theorem and Related Topics/index.html @@ -4,13 +4,13 @@ The Central Limit Theorem and Related Topics | Computing and Calculus for Applied Statistics - +
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The Central Limit Theorem and Related Topics

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- + \ No newline at end of file diff --git a/Functions/The Central Limit Theorem and Related Topics/read/index.html b/Functions/The Central Limit Theorem and Related Topics/read/index.html index 8abb61f..f69c5b5 100644 --- a/Functions/The Central Limit Theorem and Related Topics/read/index.html +++ b/Functions/The Central Limit Theorem and Related Topics/read/index.html @@ -4,7 +4,7 @@ The Central Limit Theorem and Related Topics | Computing and Calculus for Applied Statistics - + @@ -86,7 +86,7 @@ c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​x−μ​

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
- + \ No newline at end of file diff --git a/Functions/index.html b/Functions/index.html index 3a70008..ccf6491 100644 --- a/Functions/index.html +++ b/Functions/index.html @@ -4,13 +4,13 @@ Functions | Computing and Calculus for Applied Statistics - +
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Functions

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- + \ No newline at end of file diff --git a/Multivariate to Power/Estimation, Estimates and Estimators/index.html b/Multivariate to Power/Estimation, Estimates and Estimators/index.html index 707f2c1..c46bdb6 100644 --- a/Multivariate to Power/Estimation, Estimates and Estimators/index.html +++ b/Multivariate to Power/Estimation, Estimates and Estimators/index.html @@ -4,13 +4,13 @@ Estimation, Estimates and Estimators | Computing and Calculus for Applied Statistics - +
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Estimation, Estimates and Estimators

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- + \ No newline at end of file diff --git a/Multivariate to Power/Estimation, Estimates and Estimators/read/index.html b/Multivariate to Power/Estimation, Estimates and Estimators/read/index.html index 6253ce9..c251217 100644 --- a/Multivariate to Power/Estimation, Estimates and Estimators/read/index.html +++ b/Multivariate to Power/Estimation, Estimates and Estimators/read/index.html @@ -4,7 +4,7 @@ Estimation, Estimates and Estimators | Computing and Calculus for Applied Statistics - + @@ -57,7 +57,7 @@ s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​σ​ 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 72e7a9e..9bf6371 100644 --- a/Multivariate to Power/Multivariate Probability Distributions/index.html +++ b/Multivariate to Power/Multivariate Probability Distributions/index.html @@ -4,13 +4,13 @@ Multivariate Probability Distributions | Computing and Calculus for Applied Statistics - +
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Multivariate Probability Distributions

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- + \ No newline at end of file diff --git a/Multivariate to Power/Multivariate Probability Distributions/read/index.html b/Multivariate to Power/Multivariate Probability Distributions/read/index.html index fd47f9e..dc4e4e7 100644 --- a/Multivariate to Power/Multivariate Probability Distributions/read/index.html +++ b/Multivariate to Power/Multivariate Probability Distributions/read/index.html @@ -4,7 +4,7 @@ Multivariate Probability Distributions | Computing and Calculus for Applied Statistics - + @@ -76,7 +76,7 @@ c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​σ2​1​e2σ22​−(y−μ2​)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.

- + \ No newline at end of file diff --git a/Multivariate to Power/Overview/index.html b/Multivariate to Power/Overview/index.html index 0d90d70..b2d2e73 100644 --- a/Multivariate to Power/Overview/index.html +++ b/Multivariate to Power/Overview/index.html @@ -4,13 +4,13 @@ Overview | Computing and Calculus for Applied Statistics - +
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Overview

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- + \ No newline at end of file diff --git a/Multivariate to Power/Overview/read/index.html b/Multivariate to Power/Overview/read/index.html index d7b12a5..a6dfa6a 100644 --- a/Multivariate to Power/Overview/read/index.html +++ b/Multivariate to Power/Overview/read/index.html @@ -4,13 +4,13 @@ Multivariate to Power | Computing and Calculus for Applied Statistics - +
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- + \ No newline at end of file diff --git a/Multivariate to Power/Power and Sample Sizes/index.html b/Multivariate to Power/Power and Sample Sizes/index.html index 779679f..8770ed1 100644 --- a/Multivariate to Power/Power and Sample Sizes/index.html +++ b/Multivariate to Power/Power and Sample Sizes/index.html @@ -4,13 +4,13 @@ Power and Sample Sizes | Computing and Calculus for Applied Statistics - +
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Power and Sample Sizes

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- + \ No newline at end of file diff --git a/Multivariate to Power/Power and Sample Sizes/read/index.html b/Multivariate to Power/Power and Sample Sizes/read/index.html index e2d1a2b..bbf0b31 100644 --- a/Multivariate to Power/Power and Sample Sizes/read/index.html +++ b/Multivariate to Power/Power and Sample Sizes/read/index.html @@ -4,7 +4,7 @@ Power and Sample Sizes | Computing and Calculus for Applied Statistics - + @@ -733,7 +733,7 @@ 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%.

- + \ No newline at end of file diff --git a/Multivariate to Power/Some Distributions Related to Normal/index.html b/Multivariate to Power/Some Distributions Related to Normal/index.html index 75b780a..41aedeb 100644 --- a/Multivariate to Power/Some Distributions Related to Normal/index.html +++ b/Multivariate to Power/Some Distributions Related to Normal/index.html @@ -4,13 +4,13 @@ Some Distributions Related to Normal | Computing and Calculus for Applied Statistics - +
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Some Distributions Related to Normal

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- + \ No newline at end of file diff --git a/Multivariate to Power/Some Distributions Related to Normal/read/index.html b/Multivariate to Power/Some Distributions Related to Normal/read/index.html index a23073a..d5d9c6b 100644 --- a/Multivariate to Power/Some Distributions Related to Normal/read/index.html +++ b/Multivariate to Power/Some Distributions Related to Normal/read/index.html @@ -4,7 +4,7 @@ Some Distributions Related to the Normal | Computing and Calculus for Applied Statistics - + @@ -60,7 +60,7 @@ s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​Xˉ−μ​∼N(0,1), ∑(Xi−Xˉ)2σ2∼χn−12\displaystyle\sum \displaystyle\frac{(X_i-\bar{X})^2}{\sigma^2}\sim \chi_{n-1}^2.

- + \ No newline at end of file diff --git a/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/index.html b/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/index.html index 5486fdb..f66d2af 100644 --- a/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/index.html +++ b/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/index.html @@ -4,13 +4,13 @@ Test of Hypothesis, P Values and Related Concepts | Computing and Calculus for Applied Statistics - +
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Test of Hypothesis, P Values and Related Concepts

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- + \ No newline at end of file diff --git a/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/read/index.html b/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/read/index.html index 937f666..5b0bc49 100644 --- a/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/read/index.html +++ b/Multivariate to Power/Test of Hypothesis, P Values and Related Concepts/read/index.html @@ -4,7 +4,7 @@ Test of Hypothesis, P Values and Related Concepts | Computing and Calculus for Applied Statistics - + @@ -191,7 +191,7 @@ c-8,0,-12,-0.7,-12,-2z M1001 80 h400000v40h-400000z">​x−y​​

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:

But one does not say:

Similarly, if the hypothesis H0:μ1=μ2H_0: \mu_1 = \mu_2 cannot be rejected, we can say:

But we cannot say:

- + \ No newline at end of file diff --git a/Multivariate to Power/index.html b/Multivariate to Power/index.html index efecb76..0176879 100644 --- a/Multivariate to Power/index.html +++ b/Multivariate to Power/index.html @@ -4,13 +4,13 @@ Multivariate to Power | Computing and Calculus for Applied Statistics - +
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Multivariate to Power

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- + \ No newline at end of file diff --git a/Numbers to Indices/Data Vectors/index.html b/Numbers to Indices/Data Vectors/index.html index da6a383..bbe3ea9 100644 --- a/Numbers to Indices/Data Vectors/index.html +++ b/Numbers to Indices/Data Vectors/index.html @@ -4,13 +4,13 @@ Data Vectors | Computing and Calculus for Applied Statistics - +
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Data Vectors

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- + \ No newline at end of file diff --git a/Numbers to Indices/Data Vectors/read/index.html b/Numbers to Indices/Data Vectors/read/index.html index 5b78e67..8bf5cf6 100644 --- a/Numbers to Indices/Data Vectors/read/index.html +++ b/Numbers to Indices/Data Vectors/read/index.html @@ -4,7 +4,7 @@ Data Vectors | Computing and Calculus for Applied Statistics - + @@ -72,7 +72,7 @@ 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
- + \ No newline at end of file diff --git a/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/index.html b/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/index.html index 924fc0d..28c2d28 100644 --- a/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/index.html +++ b/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/index.html @@ -4,13 +4,13 @@ Discrete Random Variables and the Binomial Distribution | Computing and Calculus for Applied Statistics - +
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Discrete Random Variables and the Binomial Distribution

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- + \ No newline at end of file diff --git a/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/read/index.html b/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/read/index.html index f6edf71..529e847 100644 --- a/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/read/index.html +++ b/Numbers to Indices/Discrete Random Variables and the Binomial Distribution/read/index.html @@ -4,7 +4,7 @@ Discrete Random Variables and the Binomial Distribution | Computing and Calculus for Applied Statistics - + @@ -71,7 +71,7 @@ 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)

- + \ No newline at end of file diff --git a/Numbers to Indices/Functions/index.html b/Numbers to Indices/Functions/index.html index 98eac23..332ea15 100644 --- a/Numbers to Indices/Functions/index.html +++ b/Numbers to Indices/Functions/index.html @@ -4,13 +4,13 @@ Functions | Computing and Calculus for Applied Statistics - +
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Functions

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- + \ No newline at end of file diff --git a/Numbers to Indices/Functions/read/index.html b/Numbers to Indices/Functions/read/index.html index f06cc4d..06bf5ff 100644 --- a/Numbers to Indices/Functions/read/index.html +++ b/Numbers to Indices/Functions/read/index.html @@ -4,7 +4,7 @@ Functions | Computing and Calculus for Applied Statistics - + @@ -16,7 +16,7 @@ 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}

- + \ No newline at end of file diff --git a/Numbers to Indices/Indices and the apply Commands in R/index.html b/Numbers to Indices/Indices and the apply Commands in R/index.html index a165edb..ab7db82 100644 --- a/Numbers to Indices/Indices and the apply Commands in R/index.html +++ b/Numbers to Indices/Indices and the apply Commands in R/index.html @@ -4,13 +4,13 @@ Indices and the apply Commands in R | Computing and Calculus for Applied Statistics - +
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Indices and the apply Commands in R

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- + \ No newline at end of file diff --git a/Numbers to Indices/Indices and the apply Commands in R/read/index.html b/Numbers to Indices/Indices and the apply Commands in R/read/index.html index c79e133..be0e95c 100644 --- a/Numbers to Indices/Indices and the apply Commands in R/read/index.html +++ b/Numbers to Indices/Indices and the apply Commands in R/read/index.html @@ -4,7 +4,7 @@ Indices and the Apply Commands in R | Computing and Calculus for Applied Statistics - + @@ -18,7 +18,7 @@ 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"
- + \ No newline at end of file diff --git a/Numbers to Indices/More on Algebra/index.html b/Numbers to Indices/More on Algebra/index.html index e98628e..7999f67 100644 --- a/Numbers to Indices/More on Algebra/index.html +++ b/Numbers to Indices/More on Algebra/index.html @@ -4,13 +4,13 @@ More on Algebra | Computing and Calculus for Applied Statistics - +
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More on Algebra

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- + \ No newline at end of file diff --git a/Numbers to Indices/More on Algebra/read/index.html b/Numbers to Indices/More on Algebra/read/index.html index a49c78e..6cdf044 100644 --- a/Numbers to Indices/More on Algebra/read/index.html +++ b/Numbers to Indices/More on Algebra/read/index.html @@ -4,7 +4,7 @@ More on Algebra | Computing and Calculus for Applied Statistics - + @@ -20,7 +20,7 @@ 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}

- + \ No newline at end of file diff --git a/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/index.html b/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/index.html index 16e8e2a..7149ed3 100644 --- a/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/index.html +++ b/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/index.html @@ -4,13 +4,13 @@ Numbers, Arithmetic and Basic Algebra | Computing and Calculus for Applied Statistics - +
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Numbers, Arithmetic and Basic Algebra

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- + \ No newline at end of file diff --git a/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/read/index.html b/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/read/index.html index df452cc..a1e3550 100644 --- a/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/read/index.html +++ b/Numbers to Indices/Numbers, Arithmetic and Basic Algebra/read/index.html @@ -4,7 +4,7 @@ Numbers, Arithmetic and Basic Algebra | Computing and Calculus for Applied Statistics - + @@ -198,7 +198,7 @@ c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​ 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.

- + \ No newline at end of file diff --git a/Numbers to Indices/Overview/index.html b/Numbers to Indices/Overview/index.html index 7817565..019ff63 100644 --- a/Numbers to Indices/Overview/index.html +++ b/Numbers to Indices/Overview/index.html @@ -4,13 +4,13 @@ Overview | Computing and Calculus for Applied Statistics - +
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Overview

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Polynomials

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- + \ No newline at end of file diff --git a/Numbers to Indices/Polynomials/read/index.html b/Numbers to Indices/Polynomials/read/index.html index 495da11..9412fe8 100644 --- a/Numbers to Indices/Polynomials/read/index.html +++ b/Numbers to Indices/Polynomials/read/index.html @@ -4,7 +4,7 @@ Polynomials | Computing and Calculus for Applied Statistics - + @@ -110,7 +110,7 @@ s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">​

- + \ No newline at end of file diff --git a/Numbers to Indices/Simple Data Analysis in R/index.html b/Numbers to Indices/Simple Data Analysis in R/index.html index 132feb4..be20023 100644 --- a/Numbers to Indices/Simple Data Analysis in R/index.html +++ b/Numbers to Indices/Simple Data Analysis in R/index.html @@ -4,13 +4,13 @@ Simple Data Analysis in R | Computing and Calculus for Applied Statistics - +
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Simple Data Analysis in R

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- + \ No newline at end of file diff --git a/Numbers to Indices/Simple Data Analysis in R/read/index.html b/Numbers to Indices/Simple Data Analysis in R/read/index.html index 2d729c8..0332bd7 100644 --- a/Numbers to Indices/Simple Data Analysis in R/read/index.html +++ b/Numbers to Indices/Simple Data Analysis in R/read/index.html @@ -4,7 +4,7 @@ Simple Data Analysis in R | Computing and Calculus for Applied Statistics - + @@ -31,7 +31,7 @@ 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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Numbers to Indices

+ \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/index.html b/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/index.html index f660666..2b0abb2 100644 --- a/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/index.html +++ b/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/index.html @@ -4,13 +4,13 @@ Independence, Expectations and the Moment-Generating Function | Computing and Calculus for Applied Statistics - +
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Independence, Expectations and the Moment-Generating Function

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- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/read/index.html b/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/read/index.html index 32b94d0..dfe47d2 100644 --- a/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/read/index.html +++ b/Vectors to Some Regression Topics/Independence, Expectations and the Moment-Generating Function/read/index.html @@ -4,7 +4,7 @@ Independence, Expectations and the Moment Generating Function | Computing and Calculus for Applied Statistics - + @@ -13,7 +13,7 @@ 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.

- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Multivariate Calculus/index.html b/Vectors to Some Regression Topics/Multivariate Calculus/index.html index 75a0116..c2528c9 100644 --- a/Vectors to Some Regression Topics/Multivariate Calculus/index.html +++ b/Vectors to Some Regression Topics/Multivariate Calculus/index.html @@ -4,13 +4,13 @@ Multivariate Calculus | Computing and Calculus for Applied Statistics - +
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Multivariate Calculus

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- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Multivariate Calculus/read/index.html b/Vectors to Some Regression Topics/Multivariate Calculus/read/index.html index 13027d4..4711946 100644 --- a/Vectors to Some Regression Topics/Multivariate Calculus/read/index.html +++ b/Vectors to Some Regression Topics/Multivariate Calculus/read/index.html @@ -4,7 +4,7 @@ Multivariate Calculus | Computing and Calculus for Applied Statistics - + @@ -19,7 +19,7 @@ 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}|

- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/index.html b/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/index.html index f94b9ae..7646aef 100644 --- a/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/index.html +++ b/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/index.html @@ -4,13 +4,13 @@ Notes and Examples, The Linear Model | Computing and Calculus for Applied Statistics - +
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Notes and Examples, The Linear Model

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- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/read/index.html b/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/read/index.html index 66b3566..2b9722b 100644 --- a/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/read/index.html +++ b/Vectors to Some Regression Topics/Notes and Examples, The Linear Model/read/index.html @@ -4,7 +4,7 @@ Notes and Examples: The Linear Model | Computing and Calculus for Applied Statistics - + @@ -47,7 +47,7 @@ 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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Overview

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Ranks and Determinants

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- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Ranks and Determinants/read/index.html b/Vectors to Some Regression Topics/Ranks and Determinants/read/index.html index a5a2ad6..2bef4ae 100644 --- a/Vectors to Some Regression Topics/Ranks and Determinants/read/index.html +++ b/Vectors to Some Regression Topics/Ranks and Determinants/read/index.html @@ -4,7 +4,7 @@ Ranks and Determinants | Computing and Calculus for Applied Statistics - + @@ -14,7 +14,7 @@ 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 :

Details​

Suppose AA is an n×nn\times n matrix. Then the following are truths:

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Some Notes on Matrices and Linear Operators

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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.

- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/The Gamma Distribution/index.html b/Vectors to Some Regression Topics/The Gamma Distribution/index.html index 28a8287..4198a7f 100644 --- a/Vectors to Some Regression Topics/The Gamma Distribution/index.html +++ b/Vectors to Some Regression Topics/The Gamma Distribution/index.html @@ -4,13 +4,13 @@ The Gamma Distribution | Computing and Calculus for Applied Statistics - +
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The Gamma Distribution

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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 index a09b5b8..20cef74 100644 --- 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 @@ -4,13 +4,13 @@ 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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- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/read/index.html b/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/read/index.html index a916011..db920de 100644 --- a/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/read/index.html +++ b/Vectors to Some Regression Topics/The Multivariate Normal Distribution and Related Topics/read/index.html @@ -4,7 +4,7 @@ The Multivariate Normal Distribution and Related Topics | Computing and Calculus for Applied Statistics - + @@ -69,7 +69,7 @@ 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.

- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Vector and Matrix Operations/index.html b/Vectors to Some Regression Topics/Vector and Matrix Operations/index.html index 8ea3012..5792e85 100644 --- a/Vectors to Some Regression Topics/Vector and Matrix Operations/index.html +++ b/Vectors to Some Regression Topics/Vector and Matrix Operations/index.html @@ -4,13 +4,13 @@ Vector and Matrix Operations | Computing and Calculus for Applied Statistics - +
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Vector and Matrix Operations

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- + \ No newline at end of file diff --git a/Vectors to Some Regression Topics/Vector and Matrix Operations/read/index.html b/Vectors to Some Regression Topics/Vector and Matrix Operations/read/index.html index 0ebcc9c..3047aa2 100644 --- a/Vectors to Some Regression Topics/Vector and Matrix Operations/read/index.html +++ b/Vectors to Some Regression Topics/Vector and Matrix Operations/read/index.html @@ -4,7 +4,7 @@ Vectors and Matrix Operations | Computing and Calculus for Applied Statistics - + @@ -17,7 +17,7 @@ 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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-

Intro

This is a landing page for the CCAS course. -Here you will find all the documentation needed for this course.

Best of luck!

Community
Copyright © 2023 CCAS Team
- +

Introduction

This is a landing page for the CCAS (Computing and Calculus for Advanced Statistics) course. +Here you will find all the documentation needed for this course. +It is meant to be used by teachers, trainers, students and hobbyists who want to learn about topics on calculus and statistics.

The course is structured in chapters, each with their own sections. +Each section presents a particular topic, rich in examples. +There is a sizeable focus on the use of the R programming language for statistical computing for demonstrating the topics in a practical manner.

Chapters are:

Licensing and Contributing​

The CCAS contents are open educational resources (OER), part of the Open Education Hub project; +they are hosted on GitHub, licensed under CC BY-SA 4.0 and BSD 3-Clause.

If you find an issue or want to contribute, follow the contribution guidelines on GitHub.

Community
Copyright © 2023 CCAS Team
+ \ No newline at end of file diff --git a/markdown-page/index.html b/markdown-page/index.html index 16e589e..78c6632 100644 --- a/markdown-page/index.html +++ b/markdown-page/index.html @@ -4,13 +4,13 @@ Markdown page example | Computing and Calculus for Applied Statistics - +

Markdown page example

You don't need React to write simple standalone pages.

Community
Copyright © 2023 CCAS Team
- + \ No newline at end of file