University of Alberta Guide/STAT/222/LaTeX Version

University of Alberta Guide > STAT > 222

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\title{STAT 222\\Survival Guide} \author{By Pat Sissons} \date{Last Updated on \today}

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\part*{DON'T\\PANIC}

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\section*{Preamble} This guide is designed to help anyone who is completely lost in STAT 222. I wrote this because I am currently taking STAT 222 and I have no clue what the hell is going on. So my goal here is to pass on any knowledge that I can in terms or words that should make sense rather than a bunch of STAT 222 jargon. I only plan on filling in what I research, which is usually what I don't already know. Hopefully, when I am done with this guide, someone else can take over and continue adding more and more. Eventually, this survival guide may turn into a replacement for the real book.\\ Make note that this guide does not include any information on discrete random variables since they are not part of the STAT 222 course material.\\

\chapter{Uniform RV's} Uniform Random Variables are used where each outcome of an event is equally probable. For $p_{U}(u) = P(U \leq u)$ where $U$ is some $[a, b]$-Uniform Random Variable, $p_{U}(u)$ will always be zero unless $(a \leq u \leq b)$. When $(a \leq u \leq b)$, then $p_{U}(u) = \dfrac{1}{b-a}$.\\ \section*{Notable Equations} \begin{align*} \int_{-\infty}^{\infty}f_{X}(x)\delta x& = 1\\ \end{align*}

\chapter{Independence and Conditional Expectations} \begin{figure}[h] $P(X_{1} \leq x_{1}, ..., X_{d} \leq x_{d}) = P(X_{1} \leq x_{1}) \cdot \cdot \cdot P(X_{d} \leq x_{d})$\\ \text{The same goes for }$F_{X_{1...d}}(x_{1...d})$, $f_{X_{1...d}}(x_{1...d})$, $E\left[X_{1...d}\right]$ \caption{If RV's are independent, then you can multiply them together} \end{figure} \begin{figure}[h] $M_{X_{1} + ... + X_{d}}(t) = M_{X_{1}}(t) \cdot\cdot\cdot M_{X_{d}}(t) = E\left[e^{tX_{1}} \cdot\cdot\cdot e^{tX_{d}}\right]$ \caption{Moments, however, are a little different} \end{figure}

\part{STAT Formulas and Functions}

\section*{General Formulas} \begin{align*} R_{T}(t)& = P(T > t) = 1 - P(T \leq t) = 1 - F_{T}(t)& \forall t \geq 0\\ F_{T}(t)& = P(T \leq t) = 1 - P(T > t) = 1 - R_{T}(t)& \forall t \geq 0\\ F_{X_{1, ..., d}}(x_{1, ..., d})& = P(X_{1} \leq x_{1}, ..., X_{d} \leq x_{d})\\ R_{X_{1, ..., d}}(x_{1, ..., d})& = P(X_{1} > x_{1}, ..., X_{d} > x_{d})\\ P(X_{1} \leq x_{1}, ..., X_{d} \leq x_{d})& = P(X_{1} \leq x_{1}) \cdot \cdot \cdot P(X_{d} \leq x_{d})& \text{(By Independence)}\\ P(a \leq X < b)& = P(a \leq X \leq b) = P(a < X \leq b) = P(a < X < b)\\ P(X \leq x | Y \leq y)& = ?\\ F_{X | Y}(x | y)& = ?\\ f_{X | Y}(x | y)& = ?\\ E\left[X | Y\right]& = ?\\ P(T = t)& = 0\\ R_{X}(0)& = 1\\ F_{X}(0)& = 0\\ F_{X}(x)& = \int_{-\infty}^{x}f_{X}(y)\delta y\\ f_{X}(x)& = \frac{\delta}{\delta x}F_{X}(x)\\ P(a \leq X \leq b)& = \int_{a}^{b}f_{X}(x)\delta x\\ h_{T}(t)& = \int_{0}^{t}f_{T}(x)\delta x\\ E\left[X\right]& = \int_{-\infty}^{\infty}xf_{X}(x)\delta x\\ E\left[g(X)\right]& = \int_{-\infty}^{\infty}g(x)f_{X}(x)\delta x\\ E\left[g(Y) | X = x \right]& = \int_{-\infty}^{\infty}g(x)f_{Y | X}(y | x)\delta x\\ E\left[X^{\alpha}\right]& = \alpha \int_{0}^{\infty}t^{\alpha-1}P\left(X > t\right)\delta t\\ Cov(X)& = (x - EX)(x - EX)^{T}\\ Cov(X)& = \begin{bmatrix} (x_{1} - EX_{1})^{2} & \hdots & (x_{1} - EX_{1})(x_{d} - EX_{d})\\ \vdots & \ddots & \vdots\\ (x_{d} - EX_{d})(x_{1} - EX_{1}) & \hdots & (x_{d} - EX_{d})^{2}\\ \end{bmatrix}\\ \end{align*}

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\section*{Probability Density Functions ($pdf$)} \begin{align*} \text{Uniform}& : [a,b]-U& f_{X}(x) & = \begin{cases} \dfrac{1}{b-a}& a < x < b\\ 0& else\\ \end{cases}\\ \text{Exponential}& : \lambda-Exp& f_{X}(x) & = \begin{cases} \lambda e^{-\lambda x}& \lambda, x \geq 0\\ 0& x < 0\\ \end{cases}\\ \text{Gamma}& : (\alpha, \beta)-Gamma& f_{X}(x) & = \dfrac{x^{\alpha-1}e^{-\frac{x}{\beta}}}{\beta^{\alpha}\Gamma(\alpha)}& \alpha,\beta,x > 0\\ \text{Weibull}& : (\alpha, \beta)-Weibull& f_{X}(x) & = \left(\frac{\alpha}{\beta}\right)\left(\frac{x}{\beta}\right)^{\alpha-1}e^{-\left(\frac{x}{\beta}\right)^{\alpha}}& \alpha,\beta,x > 0\\ \text{Normal}& : (\mu, \sigma^{2})& f_{X}(x) & = \dfrac{1}{\sqrt{2\pi}\rho}e^{\left(-\frac{\left(x-\mu\right)^{2}}{2\rho^{2}}\right)}& \rho > 0, x,\mu \in \mathbb{R}\\ \text{Binomial}& : ?& p(x) & = \binom{n}{x}p^{x}\left(1-p\right)^{n-x}& x=0,1,2,...n\\ \text{Geometric}& : ?& p(x) & = p\left(1-p\right)^{x-1}& x=1,2,...,\infty\\ \text{Poisson}& : ?& p(x) & = e^{-\mu}\frac{\mu^{x}}{x!}& x=0,1,2,...,\infty\\ \end{align*}

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\section*{Cumulative Density Functions ($cdf$)} \begin{align*} \text{Uniform}& : [a,b]-\text{Uniform}& F_{X}(x) & = \begin{cases} 0& x < a\\ \dfrac{x-a}{b-a}& a \leq x < b\\ 1& x \geq b\\ \end{cases}\\ \text{Exponential}& : \lambda-\text{Exp}& F_{X}(x) & = \begin{cases} 1 - e^{-\lambda x}& \lambda, x \geq 0\\ 0& x < 0\\ \end{cases}\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& F_{X}(x) & = \dfrac{\gamma\left(\alpha,\frac{x}{\beta}\right)}{\Gamma(\alpha)}\\ & & \gamma\left(\alpha, x\right)& = \int_{0}^{x}t^{a-1}e^{-t}\delta t\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& F_{X}(x) & = 1 - e^{-\left(x/\beta\right)^{\alpha}}\\ \end{align*}

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\section*{Hazard Functions} \begin{align*} \text{Uniform}& : [a,b]-\text{Uniform}& h_{X}(x) & = ???\\ \text{Exponential}& : \lambda-\text{Exp}& h_{X}(x) & = ???\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& h_{X}(x) & = ???\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& h_{X}(x) & = \left(\frac{\alpha}{\beta}\right)\left(\frac{x}{\beta}\right)^{\left(\alpha - 1\right)}\\ \end{align*}

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\section*{Moment Generating Functions} In general, $M_{X}(u) = E\left[e^{uX}\right] = \int_{-\infty}^{\infty}e^{ux}f_{X}(x)\delta x = 1 + uE[X] + \dfrac{u^{2}E\left[X^{2}\right]}{2!} + ... + \dfrac{u^{n}E\left[X^{n}\right]}{n!}$\\ Therefore, $\dfrac{\delta^{k}}{\delta u^{k}}M_{X}(u)\vert_{u=0} = E\left[X^{k}\right]$ for $k = 0, 1, 2, ...$\\ $M_{X_{1} + ... + X_{d}}(t) = M_{X_{1}}(t) \cdot\cdot\cdot M_{X_{d}}(t) = E\left[e^{tX_{1}} \cdot\cdot\cdot e^{tX_{d}}\right]$\\ $M_{aX + b}(u) = e^{bu}M_{X}(au)$\\ \begin{align*} \text{Uniform}& : [a,b]-\text{Uniform}& M_{X}(t)& = \dfrac{e^{tb}-e^{ta}}{t\left(b-a\right)}\\ \text{Exponential}& : \lambda-\text{Exp}& M_{X}(t)& = \dfrac{\lambda}{\lambda - t}\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& M_{X}(t)& = \left(\dfrac{1}{1 - \beta t}\right)^{\alpha}& \text{for } t < 1/\beta\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& M_{X}(t)& = ???\\ \end{align*}

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\section*{Mean Functions} \begin{align*} \text{Uniform}& : [a,b]-U& = \dfrac{a+b}{2}\\ \text{Exponential}& : \lambda-\text{Exp}& = ???\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& = \alpha\beta\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& = ???\\ \end{align*}

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\section*{Median Functions} \begin{align*} \text{Uniform}& : [a,b]-U& = \dfrac{a+b}{2}\\ \text{Exponential}& : \lambda-\text{Exp}& = ???\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& = ???\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& = ???\\ \end{align*}

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\section*{Variance Functions} \begin{align*} \text{Uniform}& : [a,b]-U& = \dfrac{\left(b-a\right)^{2}}{12}\\ \text{Exponential}& : \lambda-\text{Exp}& = ???\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& = \alpha\beta^{2}\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& = ???\\ \end{align*}

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\section*{Entropy Functions} \begin{align*} \text{Uniform}& : [a,b]-U& = ln\left(b-a\right)\\ \text{Exponential}& : \lambda-\text{Exp}& = ???\\ \text{Gamma}& : (\alpha, \beta)-\text{Gamma}& = ???\\ \text{Weibull}& : (\alpha, \beta)-\text{Weibull}& = ???\\ \end{align*}

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\part{Solved Problems}

\section*{Winter 2006 Assignment 1} Combining RVs, Gamma Distributions, Weibull Distributions\\ 3.2.3, 3.2.6 (mistake found - Please use [a\_i,b\_i] instead of [0,1]), 3.2.7, 3.3.3 (ans: 2ln(4/3)), 3.3.4 (hint: see Stat 221 Notes on line), 3.3.5, 3.3.6, 3.4.1, 3.4.4, 3.4.5\\

\section*{Winter 2006 Assignment 2} Combining RVs, Gamma Distributions, Weibull Distributions\\ 3.5.2, 3.5.4 (ans: 0.696735), 3.5.5, 3.5.6, 3.6.3, 3.6.4 (ans: 0.062), 3.6.9, 3.6.11, 3.7.1, 3.7.4 (ans: 0.696665)\\

\section*{Winter 2006 Assignment 3} Hazard Rates, Laws of Large Numbers, Regression, Moment Generating Function\\ 3.8.2, 3.8.6, 3.8.7, 4.2.1, 4.2.2, 4.2.3 (There are typos in the errors. I always wanted to say that.), 4.2.4, 4.3.4, 4.3.5, 4.3.6\\

\section*{Examples on pdf and calculus}

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