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cookbook.tex
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% ----------------------------------------------------------------------------
%
% Probability and Statistics
% Cookbook
%
% ----------------------------------------------------------------------------
%
% Copyright © Matthias Vallentin <[email protected]>, 2011
%
%
\documentclass[landscape]{article}
\usepackage{array}
\usepackage{amsmath,amssymb}
\usepackage{booktabs}
\usepackage{caption}
\usepackage[nodayofweek]{datetime}
\usepackage{environ}
\usepackage{float}
\usepackage{enumitem}
\usepackage{fancyhdr}
\usepackage[landscape,margin=13mm,footskip=1pt,includefoot]{geometry}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{multicol}
\usepackage{rotating}
\usepackage{tikz}
\usepackage{translator}
\usepackage{url}
\usepackage{xspace}
% Alias for \translate.
\newcommand{\T}[1]{\translate{#1}}
% Language configuration.
\input{./config}
\usedictionary{probstat}
% Probability and Statistics LaTeX shortcuts. (Not part of the repository.)
\input{probstat}
% TikZ tweaks
\usetikzlibrary{arrows,shapes}
\usetikzlibrary{decorations.pathreplacing}
\tikzstyle{every picture}+=[remember picture]
\tikzstyle{na} = [baseline=-.5ex]
% Move footnotes to the bottom-right corner
\pagestyle{fancy}
\fancyhf{} % clear all header and footer fields
\fancyhead{}
%\fancyfoot[C]{\footnotesize Copyright (c) Matthias Vallentin, 2010}
\fancyfoot[R]{\footnotesize \thepage}
\renewcommand{\headrulewidth}{0pt}
% Further document tweaks.
\parindent=0pt
\setitemize{itemsep=0.2mm,parsep=1pt}
\setenumerate{itemsep=0.2mm,parsep=1pt}
% Link style (hyperref package)
\hypersetup{
colorlinks=true, % false: boxed links; true: colored links
linkcolor=red, % color of internal links
citecolor=cyan, % color of links to bibliography
filecolor=magenta, % color of file links
urlcolor=blue % color of external links
}
% Personal
\def\email{[email protected]}
\def\web{%
\url{http://matthias.vallentin.net/probability-and-statistics-cookbook/}}
% An itemize list with a title that avoids a break between title and list.
\newenvironment{titemize}[1]{
\begin{minipage}[h]{\columnwidth}
#1
\begin{itemize}
}{
\end{itemize}
\end{minipage}
}
\begin{document}
\thispagestyle{empty}
\begin{center}
\vspace*{\fill}
\textsc{\Huge \T{maintitle}\\[2ex] \huge \T{subtitle}}
\vfill
\footnotesize{
Copyright \copyright{}
\href{http://matthias.vallentin.net}{Matthias Vallentin}, \number\year\\
\texttt{\href{mailto:\email}{\email}}\\[2ex]
\today
}
\end{center}
\newpage
\thispagestyle{empty}
\begin{multicols*}{3}
{\footnotesize \T{disclaimer}}
\tableofcontents
\end{multicols*}
\newpage
\section{\T{Distribution Overview}}
\subsection{\T{Discrete Distributions}}
\begin{center}
\small
\begin{tabular}{@{}l*6{>{\begin{math}\displaystyle}c<{\end{math}}}@{}}
\toprule &&&&&& \\[-2ex]
& \text{\T{Notation}}\footnotemark
& F_X(x) & f_X(x) & \E{X} & \V{X} & M_X(s) \\[1ex]
\midrule
\T{Uniform} & \unifd & \punifd & \dunifd &
\frac{a+b}{2} & \frac{(b-a+1)^2-1}{12} &
\frac{e^{as}-e^{-(b+1)s}}{s(b-a)} \\[3ex]
\T{Bernoulli} & \bern & \pbern & \dbern &
p & p(1-p) &
1-p+pe^s \\[3ex]
\T{Binomial} & \bin & I_{1-p}(n-x,x+1) & \dbin &
np & np(1-p) &
(1-p+pe^s)^n \\[3ex]
\T{Multinomial} & \mult & & \dmult \quad \sum_{i=1}^k x_i = n&
np_i & np_i(1-p_i) &
\left( \sum_{i=0}^k p_i e^{s_i} \right)^n \\[3ex]
\T{Hypergeometric} & \hyper &
\approx \Phi\left(\displaystyle\frac{x-np}{\sqrt{np(1-p)}}\right) &
\dhyper &
\frac{nm}{N} & \frac{nm(N-n)(N-m)}{N^2(N-1)} & \\[3ex]
\T{Negative Binomial} & \nbin & \pnbin & \dnbin &
r\frac{1-p}{p} & r\frac{1-p}{p^2} &
\left(\frac{p}{1-(1-p)e^s}\right)^r \\[3ex]
\T{Geometric} & \geo &
\pgeo \quad x\in\mathbb N^+ &
\dgeo \quad x\in\mathbb N^+ &
\frac{1}{p} & \frac{1-p}{p^2} &
\frac{p}{1-(1-p)e^s} \\[3ex]
\T{Poisson} & \pois & \ppois & \dpois &
\lambda & \lambda &
e^{\lambda(e^s-1)}\\[3ex]
\bottomrule
\end{tabular}
\begin{figure}[H]
\includegraphics[scale=0.35]{figs/uniform-discrete.pdf}
\includegraphics[scale=0.35]{figs/binomial.pdf}
\includegraphics[scale=0.35]{figs/geometric.pdf}
\includegraphics[scale=0.35]{figs/poisson.pdf}
\end{figure}
\end{center}
\footnotetext{\T{Footnote about notation}}
\pagebreak
\subsection{\T{Continuous Distributions}}
\begin{center}
\small
%\newcolumntype{L}{>{\varwidth[c]{\linewidth}}l<{\endvarwidth}}
\newcolumntype{M}{>{\begin{math}\displaystyle}c<{\end{math}}}
\begin{tabular}{@{}l*6{M}@{}}
\toprule &&&&&& \\[-2ex]
& \text{\T{Notation}}
& F_X(x) & f_X(x) & \E{X} & \V{X} & M_X(s) \\[1ex]
\midrule
\T{Uniform} & \unif & \punif & \dunif &
\frac{a+b}{2} & \frac{(b-a)^2}{12} &
\frac{e^{sb}-e^{sa}}{s(b-a)} \\[3ex]
\T{Normal} & \norm &
\Phi(x)=\displaystyle\int_{-\infty}^x \phi(t)\,dt &
\phi(x)=\dnorm &
\mu & \sigma^2 &
\Exp{\mu s + \frac{\sigma^2s^2}{2}}\\[3ex]
\T{Log-Normal} & \ln\norm&
\frac{1}{2}+\frac{1}{2} \erf\left[\frac{\ln x-\mu}{\sqrt{2\sigma^2}}\right] &
\frac{1}{x\sqrt{2\pi\sigma^2}} \Exp{-\frac{(\ln x - \mu)^2}{2\sigma^2}} &
e^{\mu+\sigma^2/2} &
(e^{\sigma^2}-1) e^{2\mu+\sigma^2} &
\\[3ex]
\T{Multivariate Normal} & \mvn & &
(2\pi)^{-k/2} |\Sigma|^{-1/2} e^{-\frac{1}{2}(x-\mu)^T \Sigma^{-1}(x-\mu)} &
\mu & \Sigma &
\Exp{\mu^T s + \frac{1}{2} s^T \Sigma s}\\[3ex]
\T{Student's t} & \text{Student}(\nu)
& I_x\left( \frac{\nu}{2},\frac{\nu}{2} \right)
& \frac{\Gamma\left(\frac{\nu+1}{2}\right)}
{\sqrt{\nu\pi}\Gamma\left(\frac{\nu}{2}\right)}
\left(1+\frac{x^2}{\nu}\right)^{-(\nu+1)/2}
& 0 %\; \nu > 1
& \begin{cases}
\displaystyle\frac{\nu}{\nu-2} & \nu > 2 \\
\infty & 1 < \nu \le 2
\end{cases}
& \\[3ex]
\T{Chi-square} & \chisq &
\frac{1}{\Gamma(k/2)} \gamma\left(\frac{k}{2}, \frac{x}{2}\right) &
\frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2-1} e^{-x/2}&
k & 2k &
(1-2s)^{-k/2} \; s<1/2\\[3ex]
F & \text{F}(d_1,d_2) &
I_\frac{d_1x}{d_1x+d_2}\left(\frac{d_1}{2},\frac{d_1}{2}\right) &
\frac{\sqrt{\frac{(d_1x)^{d_1} d_2^{d_2}}{(d_1x+d_2)^{d_1+d_2}}}}
{x\mathrm{B}\left(\frac{d_1}{2},\frac{d_1}{2}\right)} &
\frac{d_2}{d_2-2} %\; d_2 > 2
& \frac{2d_2^2(d_1+d_2-2)}{d_1(d_2-2)^2(d_2-4)} %\; d_2 > 4
& \\[3ex]
\T{Exponential} & \ex & \pex & \dex &
\beta & \beta^2 &
\frac{1}{1-\beta s} \left(s < 1/\beta \right) \\[3ex]
Gamma & \gam &
\frac{\gamma(\alpha,x/\beta)}{\Gamma(\alpha)} & \dgamma &
\alpha\beta & \alpha\beta^2 &
\left( \frac{1}{1-\beta s} \right)^\alpha \left( s < 1/\beta \right)\\[3ex]
\T{Inverse Gamma} & \invgamma & \pinvgamma & \dinvgamma &
\frac{\beta}{\alpha-1} \; \alpha>1 &
\frac{\beta^2}{(\alpha-1)^2(\alpha-2)^2} \; \alpha > 2 &
\frac{2(-\beta s)^{\alpha/2}}{\Gamma(\alpha)}K_\alpha
\left( \sqrt{-4\beta s} \right)\\[3ex]
Dirichlet & \dir & & \ddir &
\frac{\alpha_i}{\sum_{i=1}^k \alpha_i} &
\frac{\E{X_i}(1-\E{X_i})}{\sum_{i=1}^k\alpha_i + 1} & \\[3ex]
Beta & \bet & I_x(\alpha,\beta)& \dbeta &
\frac{\alpha}{\alpha+\beta} &
\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} &
1+\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1}
\frac{\alpha+r}{\alpha+\beta+r} \right) \frac{s^k}{k!} \\[3ex]
Weibull & \mathrm{Weibull}(\lambda, k) & 1 - e^{-(x/\lambda)^k} & \dweibull &
\lambda \Gamma\left(1 + \frac{1}{k} \right) &
\lambda^2 \Gamma\left(1 + \frac{2}{k}\right) - \mu^2 &
\sum_{n=0}^\infty \frac{s^n \lambda^n}{n!} \Gamma\left(1+\frac{n}{k}\right)
\\[3ex]
Pareto & \mathrm{Pareto}(x_m, \alpha) &
1 - \left(\frac{x_m}{x} \right)^\alpha \; x\ge x_m &
\alpha\frac{x_m^\alpha}{x^{\alpha+1}} \quad x\ge x_m&
\frac{\alpha x_m}{\alpha-1} \; \alpha>1 &
\frac{x_m^\alpha}{(\alpha-1)^2(\alpha-2)} \; \alpha>2 &
\alpha(-x_m s)^\alpha \Gamma(-\alpha,-x_m s) \; s<0\\[3ex]
\bottomrule
\end{tabular}
\end{center}
\begin{figure}[H]
\includegraphics[scale=0.35]{figs/uniform-continuous.pdf}
\includegraphics[scale=0.35]{figs/normal.pdf}
\includegraphics[scale=0.35]{figs/lognormal.pdf}
\includegraphics[scale=0.35]{figs/student.pdf}
\includegraphics[scale=0.35]{figs/chisquare.pdf}
\includegraphics[scale=0.35]{figs/f.pdf}
\includegraphics[scale=0.35]{figs/exponential.pdf}
\includegraphics[scale=0.35]{figs/gamma.pdf}
\includegraphics[scale=0.35]{figs/invgamma.pdf}
\includegraphics[scale=0.35]{figs/beta.pdf}
\includegraphics[scale=0.35]{figs/weibull.pdf}
\includegraphics[scale=0.35]{figs/pareto.pdf}
\end{figure}
\begin{multicols*}{2}
\section{\T{Probability Theory}}
\T{Definitions}
\begin{itemize}
\item \T{Sample space} $\Omega$
\item \T{Outcome} $\omega \in \Omega$
\item \T{Event} $A \subseteq \Omega$
\item \T{Sigma-algebra} $\mathcal{A}$
\begin{enumerate}
\item $\emptyset \in \mathcal{A}$
\item $A_1,A_2,\dots, \in \mathcal{A}
\imp \bigcup_{i=1}^\infty A_i \in \mathcal{A}$
\item $A \in \mathcal{A} \imp \comp{A} \in \mathcal{A}$
\end{enumerate}
\item \T{Probability Distribution} $\prob$
\begin{enumerate}
\item $\Pr{A} \ge 0 \quad \forall A$
\item $\Pr{\Omega} = 1$
\item $\Pr{\displaystyle\bigsqcup_{i=1}^\infty A_i}
= \displaystyle\sum_{i=1}^\infty \Pr{A_i}$
\end{enumerate}
\item \T{Probability space} $(\Omega,\mathcal{A},\prob)$
\end{itemize}
\T{Properties}
\begin{itemize}
\item $\Pr{\emptyset} = 0$
\item $B = \Omega \cap B = (A \cup \comp{A}) \cap B
= (A \cap B) \cup (\comp{A} \cap B)$
\item $\Pr{\comp{A}} = 1 - \Pr{A}$
\item $\Pr{B} = \Pr{A \cap B} + \Pr{\comp{A} \cap B}$
\item $\Pr{\Omega} = 1 \qquad \Pr{\emptyset} = 0$
\item $\comp{\left(\bigcup_n A_n\right)} = \bigcap_n \comp{A_n}
\quad
\comp{\left(\bigcap_n A_n\right)} = \bigcup_n \comp{A_n}
\qquad$
\textsc{DeMorgan}
\item $\Pr{\bigcup_n A_n}
= 1 - \Pr{\bigcap_n \comp{A_n}}$
\item $\Pr{A \cup B} = \Pr{A} + \Pr{B} - \Pr{A \cap B}\\[1ex]
\imp \Pr{A \cup B} \le \Pr{A} + \Pr{B}$
\item $\Pr{A \cup B}
= \Pr{A \cap \comp{B}} + \Pr{\comp{A} \cap B} + \Pr{A \cap B}$
\item $\Pr{A \cap \comp{B}} = \Pr{A} - \Pr{A \cap B}$
\end{itemize}
\T{Continuity of Probabilities}
\begin{itemize}
\item $A_1 \subset A_2 \subset \dots \imp \limn \Pr{A_n} = \Pr{A}
\quad\text{\T{where}} A = \bigcup_{i=1}^\infty A_i$
\item $A_1 \supset A_2 \supset \dots \imp \limn \Pr{A_n} = \Pr{A}
\quad\text{\T{where}} A = \bigcap_{i=1}^\infty A_i$
\end{itemize}
\T{Independence} \ind
\[A \ind B \eqv \Pr{A \cap B} = \Pr{A}\Pr{B}\]
\T{Conditional Probability}
\[\Pr{A \giv B} = \frac{\Pr{A \cap B}}{\Pr{B}} \qquad \Pr{B} > 0\]
\T{Law of Total Probability}
\[ \Pr{B} = \sum_{i=1}^n \Pr{B|A_i}\Pr{A_i}
\qquad \Omega = \bigsqcup_{i=1}^n A_i\]
\textsc{\T{Bayes' Theorem}}
\[\Pr{A_i \giv B}
= \frac{\Pr{B \giv A_i}\Pr{A_i}}{\sum_{j=1}^n \Pr{B \giv A_j}\Pr{A_j}}
\qquad \Omega = \bigsqcup_{i=1}^n A_i\]
\T{Inclusion-Exclusion Principle}
\[\biggl|\bigcup_{i=1}^n A_i\biggr| = \sum_{r=1}^n(-1)^{r-1}
\sum_{i \le i_1 < \dots < i_r \le n}\biggl|\bigcap_{j=1}^r A_{i_j}\biggr|\]
\section{\T{Random Variables}}
\T{Random Variable}
\[X: \Omega \to \R\]
\T{Probability Mass Function}
\[f_X(x) = \Pr{X = x} = \Pr{\{\omega\in\Omega:X(\omega) = x\}}\]
\T{Probability Density Function}
\[\Pr{a \le X \le b} = \int_a^b f(x)\dx\]
\T{Cumulative Distribution Function}
\[F_X:\R \to [0,1] \qquad F_X(x) = \Pr{X \le x}\]
\begin{enumerate}
\item \T{Nondecreasing}: $x_1 < x_2 \imp F(x_1) \le F(x_2)$
\item \T{Normalized}: $\lim_{x\to -\infty} = 0$
\T{and} $\lim_{x\to \infty} = 1$
\item \T{Right-Continuous}: $\lim_{y\downarrow x} F(y) = F(x)$
\end{enumerate}
\[\Pr{a\le Y\le b \giv X=x} = \int_a^b f_{Y|X}(y\giv x) dy \qquad a \le b\]
\[ f_{Y|X}(y\giv x) = \frac{f(x,y)}{f_X(x)} \]
\T{Independence}
\begin{enumerate}
\item $\Pr{X \le x, Y \le y} = \Pr{X \le x}\Pr{Y \le y}$
\item $f_{X,Y}(x,y) = f_X(x)f_Y(y)$
\end{enumerate}
\subsection{\T{Transformations}}
\T{Transformation function}
\[Z = \transform(X)\]
\T{Discrete}
\[f_Z(z) = \Pr{\transform(X) = z} = \Pr{\{x:\transform(x) = z\}}
= \Pr{X \in \transform^{-1}(z)} = \sum_{x \in \transform^{-1}(z)} \!\!\!f(x)\]
\T{Continuous}
\[F_Z(z) = \Pr{\transform(X) \le z} = \int_{A_z} f(x) \dx \quad
\text{with } A_z = \{x:\transform(x) \le z\}\]
\T{Special case if transform strictly monotone}
\[f_Z(z)
= f_X(\transform^{-1}(z))
\left|\frac{d}{dz}\transform^{-1}(z)\right|
= f_X(x)\left|\frac{dx}{dz}\right|
= f_X(x)\frac{1}{|J|}\]
\T{The Rule of the Lazy Statistician}
\[\E{Z} = \int \transform(x) \dfx\]
\[\E{I_A(x)} = \int I_A(x) \dfx = \int_A \dfx = \Pr{X \in A}\]
\T{Convolution}
\begin{itemize}
\item $ Z:=X+Y \qquad
f_Z(z)=\displaystyle\int_{-\infty}^{\infty} f_{X,Y}(x,z-x)\,dx
\;\stackrel{X,Y \ge 0}{=}\; \int_0^z f_{X,Y}(x,z-x)\,dx$
\item $ Z:=|X-Y| \qquad
f_Z(z)=\displaystyle2\int_0^\infty f_{X,Y}(x,z+x)\,dx$
%\;\stackrel{X,Y \ge 0}{=}\; \int_0^\infty f_{X,Y}(x,z+x)\,dx$
\item $ Z:=\displaystyle\frac{X}{Y} \qquad
f_Z(z)=\displaystyle\int_{-\infty}^{\infty} |x| f_{X,Y}(x,xz)\,dx
\;\stackrel{\ind}{=}\; \int_{-\infty}^{\infty} xf_x(x)f_X(x)f_Y(xz)\,dx$
\end{itemize}
% \subsection{Joint Distribution}
% \begin{itemize}
% \item $f(x,y) = \Pr{X \le k, Y \le m)}
% = \displaystyle\int_{-\infty}^k\int_{-\infty}^m f(x,y)\,dy\,dx$
% \item $\Pr{a < X \le b, c < y \le d} = F(b,d) - F(a,d) - F(b,c) + F(a,c)$
% \item $f_X(x) = \displaystyle\int_{-\infty}^\infty f(x,y)\,dy \qquad
% f_Y(y) = \displaystyle\int_{-\infty}^\infty f(x,y)\,dx$
% \end{itemize}
% Order Statistics
% \begin{itemize}
% \item $U_i\ind U_j$ continuous \textsc{RVs} with common density $f$
% \item $X_1(\omega) < \dots < X_n(\omega)$ permuted set of $U_i$'s
% \item $X_k = $ \emph{k$^{th}$ order statistic}
% \item $X_1(\omega) = \min(U_1(\omega),\dots,U_n(\omega))$
% \item $X_n(\omega) = \max(U_1(\omega),\dots,U_n(\omega))$
% \item $R(\omega) = X_n(\omega) - X_1(\omega)$
% \end{itemize}
\section{\T{Expectation}}
\T{Definition} \T{and} \T{properties}
\begin{itemize}
\item $\E{X} = \mu_X = \displaystyle \int x \dfx =
\begin{cases}
\displaystyle\sum_x xf_X(x) & \text{X \T{discrete}} \\\\
\displaystyle\int xf_X(x) & \text{X \T{continuous}}
\end{cases}$
\item $\Pr{X=c}=1 \imp \E{c} = c$
\item $\E{cX} = c\,\E{X}$
\item $\E{X+Y} = \E{X}+\E{Y}$
\item $\E{XY} = \displaystyle\int_{X,Y} xy f_{X,Y}(x,y)\dfx\dfy$
\item $\E{\transform(Y)} \neq \transform(\E{X}) \qquad$
(cf.~\hyperref[jensen]{\textsc{Jensen} \T{inequality}})
\item $\Pr{X\ge Y} = 0 \imp \E{X}\ge\E{Y}
\wedge \Pr{X=Y}=1 \imp \E{X}=\E{Y}$
% \item $\Pr{\lvert Y\rvert\le c} = 1 \imp \E{Y}<\infty
% \wedge \lvert\E{X}\rvert\le c$
\item $\E{X} = \displaystyle\sum_{x=1}^\infty \Pr{X\ge x}$
\end{itemize}
\T{Sample mean}
\[\samplemean = \frac{1}{n}\sum_{i=1}^n X_i\]
\begin{titemize}{\T{Conditional expectation}}
\item $\E{Y\giv X=x} = \displaystyle\int y f(y\giv x)\dy$
\item $\E{X} = \E{\E{X\giv Y}}$
\item $E[\transform(X,Y)\giv X=x]
= \displaystyle\int_{-\infty}^\infty \transform(x,y)f_{Y|X}(y\giv x)\dx$
\item $\E{\transform(Y,Z)\giv X=x} =
\displaystyle\int_{-\infty}^\infty\transform(y,z)
f_{(Y,Z)|X}(y,z\giv x)\,dy\,dz$
\item $\E{Y+Z\giv X} = \E{Y\giv X} + \E{Z\giv X}$
\item $\E{\transform(X)Y\giv X} = \transform(X)\E{Y\giv X}$
\item $E[Y\giv X] = c \imp \cov{X,Y}=0$
\end{titemize}
\section{\T{Variance}}
\begin{titemize}{\T{Definition} \T{and} \T{properties}}
\item $\V{X} = \sigma_X^2 = \E{(X-\E{X})^2} = \E{X^2} - \E{X}^2$
\item $\V{\displaystyle\sum_{i=1}^n X_i} =
\displaystyle\sum_{i=1}^n \V{X_i} + 2\sum_{i\ne j}\cov{X_i,Y_j}$
% \stackrel{X_i \ind X_j}{=}\sum_{i=1}^n\V{X_i}$
\item $\V{\displaystyle\sum_{i=1}^n X_i} =
\displaystyle\sum_{i=1}^n\V{X_i} \quad$ if $X_i \ind X_j$
\end{titemize}
\T{Standard deviation}
\[\sd[X] = \sqrt{\V{X}} = \sigma_X\]
\T{Covariance}
\begin{itemize}
\item $\cov{X,Y} = \E{(X-\E{X})(Y-\E{Y})} = \E{XY}-\E{X}\E{Y}$
\item $\cov{X,a} = 0$
\item $\cov{X,X} = \V{X}$
\item $\cov{X,Y} = \cov{Y,X}$
\item $\cov{aX,bY} = ab\cov{X,Y}$
\item $\cov{X+a,Y+b} = \cov{X,Y}$
\item $\cov{\displaystyle\sumin X_i, \sumjm Y_j}
= \displaystyle\sumin\sumjm\cov{X_i, Y_j}$
\end{itemize}
\T{Correlation}
\[\corr{X,Y} = \displaystyle\frac{\cov{X,Y}}{\sqrt{\V{X}\V{Y}}}\]
\T{Independence}
\[X\ind Y \imp \corr{X,Y} = 0 \eqv \cov{X,Y} = 0 \eqv \E{XY}=\E{X}\E{Y}\]
\T{Sample variance}
\[\samplevar = \frac{1}{n-1}\sum_{i=1}^n(X_i-\samplemean)^2\]
\T{Conditional variance}
\begin{itemize}
\item $\V{Y\giv X} = \E{(Y-\E{Y\giv X})^2\giv X} =\E{Y^2\giv X}-\E{Y\giv X}^2$
\item $\V{Y} = \E{\V{Y\giv X}}+\V{\E{Y\giv X}}$
\end{itemize}
\section{\T{Inequalities}}
\textsc{Cauchy-Schwarz}
\[\E{XY}^2 \le \E{X^2}\E{Y^2}\]
\textsc{Markov}
\[\Pr{\transform(X) \ge t}\le\frac{\E{\transform(X)}}{t}\]
\textsc{Chebyshev}
\[\Pr{\lvert X-\E{X}\rvert \ge t} \le \frac{\V{X}}{t^2}\]
\textsc{Chernoff}
\[\Pr{X \ge (1+\delta)\mu}
\le \left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right) \quad \delta>-1\]
\textsc{Jensen}\label{jensen}
\[\E{\transform(X)} \ge \transform(\E{X}) \quad
\transform \text{ \T{convex}}\]
\section{\T{Distribution Relationships}}
\T{Binomial}
\begin{itemize}
\item $X_i \dist \bern \imp \displaystyle\sum_{i=1}^n X_i \dist \bin$
\item $X\dist\bin, Y\dist\bin[m,p] \imp X+Y\dist\bin[n+m,p]$
\item $\limn\bin = \pois[np] \qquad$ ($n$ \T{large}, $p$ \T{small})
\item $\limn\bin = \norm[np,np(1-p)] \qquad$
($n$ \T{large}, $p$ \T{far from 0 and 1})
\end{itemize}
\T{Negative Binomial}
\begin{itemize}
\item $ X\dist \nbin[1,p] = \geo $
\item $ X\dist \nbin[r,p] = \sum_{i=1}^r \geo $
\item $X_i\dist \nbin[r_i,p] \imp \sum X_i\dist \nbin[\sum r_i,p] $
\item $X\dist \nbin[r,p].\; Y\dist \bin[s+r,p] \imp \Pr{X\le s} = \Pr{Y\ge r}$
\end{itemize}
\T{Poisson}
\begin{itemize}
\item $X_i\dist\pois[\lambda_i] \wedge X_i \ind X_j
\imp \displaystyle\sumin X_i \dist \pois[\displaystyle\sumin \lambda_i]$
\item $X_i\dist\pois[\lambda_i] \wedge X_i \ind X_j
\imp X_i\,\left|\displaystyle\sumjn X_j\right. \dist
\bin[\displaystyle\sumjn X_j,\displaystyle\frac{\lambda_i}{\sumjn\lambda_j}]$
\end{itemize}
Exponential
\begin{itemize}
% \item $\forall n \in \mathbb N^+: X_i\dist\ex{\lambda}
\item $X_i\dist\ex \wedge X_i \ind X_j
\imp \displaystyle\sumin X_i\dist \gam[n,\beta]$
\item \T{Memoryless property}: $\Pr{X>x+y\giv X>y}=\Pr{X>x}$
\end{itemize}
\T{Normal}
\begin{itemize}
\item $X\dist \norm[\mu,\sigma^2]
\imp \left(\frac{X-\mu}{\sigma}\right)\dist\norm[0,1] $
\item $X\dist \norm[\mu,\sigma^2] \wedge Z = aX+b
\imp Z\dist\norm[a\mu+b,a^2\sigma^2] $
\item $X\dist \norm[\mu_1,\sigma_1^2] \wedge Y\dist\norm[\mu_2,\sigma_2^2]
\imp X+Y\,\dist \norm[\mu_1+\mu_2,\sigma_1^2 + \sigma_2^2] $
\item $X_i\dist\norm[\mu_i,\sigma_i^2]
\imp \sum_i X_i \dist \norm[\sum_i\mu_i,\sum_i\sigma_i^2]$
\item $\Pr{a < X \le b}= \Phi\left(\frac{b-\mu}{\sigma}\right)
- \Phi\left(\frac{a-\mu}{\sigma}\right) $
\item $\Phi(-x) = 1 - \Phi(x) \qquad \phi'(x) = -x\phi(x) \qquad
\phi''(x) = (x^2-1)\phi(x)$
\item \T{Upper quantile of} $\norm[0,1]$: $z_{\alpha} = \Phi^{-1}(1-\alpha)$
\end{itemize}
Gamma
\begin{itemize}
\item $X\dist\gam \eqv X/\beta \dist\gam[\alpha,1]$
\item $\gam\dist \sum_{i=1}^\alpha\ex$
\item $X_i\dist\gam[\alpha_i,\beta] \wedge X_i \ind X_j \imp
\sum_i X_i\dist \gam[\sum_i \alpha_i,\beta]$
\item $\displaystyle\frac{\Gamma(\alpha)}{\lambda^\alpha}
= \displaystyle\int_0^\infty x^{\alpha-1} e^{-\lambda x} \dx$
\end{itemize}
Beta
\begin{itemize}
\item $\displaystyle
\frac{1}{\text{B}(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}
= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}
x^{\alpha-1}(1-x)^{\beta-1} $
\item $\E{X^k}
= \displaystyle\frac{\text{B}(\alpha+k,\beta)}{\text{B}(\alpha,\beta)}
= \displaystyle\frac{\alpha+k-1}{\alpha+\beta+k-1}\E{X^{k-1}}$
\item $\bet[1,1] \dist \unif[0,1]$
\end{itemize}
\section{\T{Probability and Moment Generating Functions}}
\begin{itemize}
\item $G_X(t) = \E{t^X} \qquad |t| < 1$
\item $M_X(t) = G_X(e^t) = \E{e^{Xt}}
= \E{ \displaystyle\sum_{i=0}^\infty \frac{(Xt)^i}{i!}}
= \displaystyle\sum_{i=0}^\infty \frac{\E{X^i}}{i!}\cdot t^i$
\item $\Pr{X=0} = G_X(0)$
\item $\Pr{X=1}=G_X'(0)$
\item $\Pr{X=i} = \displaystyle\frac{G_X^{(i)}(0)}{i!}$
\item $\E{X} = G_X'(1^-)$
\item $\E{X^k} = M_X^{(k)}(0)$
\item $\E{\displaystyle\frac{X!}{(X-k)!}} = G_X^{(k)}(1^-)$
\item $\V{X} = G_X''(1^-) + G_X'(1^-)
- \left(G_X'(1^-)\right)^2$
\item $G_X(t) = G_Y(t) \imp X \stackrel{d}{=} Y$
\end{itemize}
\section{\T{Multivariate Distributions}}
\subsection{\T{Standard Bivariate Normal}}
\T{Let} $X,Y\dist\norm[0,1] \wedge X\ind Z$ \T{where}
$Y = \rho X + \sqrt{1-\rho^2}Z$\\
\T{Joint density}
\[
f(x,y) = \frac{1}{2 \pi \sqrt{1-\rho^2}}
\Exp{-\frac{x^2 + y^2 - 2\rho x y}{2 (1-\rho^2)}}
\]
\T{Conditionals}
\[
(Y\giv X=x) \dist \norm[\rho x,1-\rho^2] \qquad\text{and}\qquad
(X\giv Y=y) \dist \norm[\rho y,1-\rho^2]
\]
\T{Independence}
\[X \ind Y \eqv \rho = 0\]
\subsection{\T{Bivariate Normal}}
% - http://www.athenasc.com/Bivariate-Normal.pdf
% - http://mathworld.wolfram.com/BivariateNormalDistribution.html
\T{Let} $X\dist\norm[\mu_x,\sigma_x^2]$
\T{and} $Y\dist\norm[\mu_y,\sigma_y^2]$.
\[f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}}
\Exp{-\frac{z}{2 (1-\rho^2)}}\]
\[ z =
\left[
\left(\frac{x-\mu_x}{\sigma_x}\right)^2
+ \left(\frac{y-\mu_y}{\sigma_y}\right)^2
- 2\rho\left(\frac{x-\mu_x}{\sigma_x}\right)
\left(\frac{y-\mu_y}{\sigma_y}\right)
\right]
\]
\T{Conditional mean and variance}
\[\E{X\giv Y} = \E{X} + \rho\frac{\sigma_X}{\sigma_Y}(Y-\E{Y})\]
\[\V{X\giv Y} = \sigma_X \sqrt{1-\rho^2}\]
\subsection{\T{Multivariate Normal}}
\T{Covariance matrix} $\Sigma$ \quad (\T{Precision matrix} $\Sigma^{-1}$)
\[\Sigma =
\begin{pmatrix}
\V{X_1} & \cdots & \cov{X_1,X_k} \\
\vdots & \ddots & \vdots \\
\cov{X_k,X_1} & \cdots & \V{X_k}
\end{pmatrix}\]
\T{If} $X \dist \norm[\mu,\Sigma]$,
\[f_X(x) = (2\pi)^{-n/2} \left|\Sigma\right|^{-1/2}
\Exp{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)} \]
\T{Properties}
\begin{itemize}
\item $Z \dist \norm[0,1] \wedge X = \mu+\Sigma^{1/2}Z
\imp X \dist \norm[\mu,\Sigma]$
\item $X \dist \norm[\mu,\Sigma] \imp \Sigma^{-1/2}(X-\mu) \dist \norm[0,1]$
\item $X \dist \norm[\mu,\Sigma] \imp AX \dist \norm[A\mu, A\Sigma A^T]$
\item $X \dist \norm[\mu,\Sigma] \wedge \|a\| = k
\imp a^TX \dist \norm[a^T\mu, a^T\Sigma a]$
\end{itemize}
\section{\T{Convergence}}
\T{Convergence Introduction}\\
\T{Types of Convergence}
\begin{enumerate}
\item \T{In distribution}: $X_n \dconv X$
\[\limn F_n(t) = F(t) \qquad
\forall t \text{ \T{where} } F \text{ \T{continuous}}\]
\item \T{In probability}: $X_n \pconv X$
\[(\forall \varepsilon > 0) \;
\lim_{n\to\infty} \Pr{|X_n -X| > \varepsilon} = 0\]
\item \T{Almost surely}: $X_n \asconv X$
\[\Pr{\limn X_n=X} = \Pr{\omega\in\Omega: \limn X_n(\omega)=X(\omega)}=1\]
\item \T{In quadratic mean}: $X_n \qmconv X$
\[\lim_{n\to\infty} \E{(X_n - X)^2} = 0\]
\end{enumerate}
\T{Relationships}
\begin{itemize}
\item $X_n \qmconv X \imp X_n \pconv X \imp X_n \dconv X$
\item $X_n \asconv X \imp X_n \pconv X$
\item $X_n \dconv X \wedge (\exists c \in \R) \; \Pr{X=c} = 1
\imp X_n \pconv X$
\item $X_n \pconv X \wedge Y_n \pconv Y
\imp X_n + Y_n \pconv X + Y$
\item $X_n \qmconv X \wedge Y_n \qmconv Y
\imp X_n + Y_n \qmconv X + Y$
\item $X_n \pconv X \wedge Y_n \pconv Y
\imp X_nY_n \pconv XY$
\item $X_n \pconv X \imp \transform(X_n) \pconv \transform(X)$
\item $X_n \dconv X \imp \transform(X_n) \dconv \transform(X)$
\item $X_n \qmconv b \eqv \lim_{n\to\infty} \E{X_n}=b
\wedge \lim_{n\to\infty} \V{X_n} = 0$
\item $X_1,\dots,X_n\; \iid \wedge \E{X}=\mu \wedge \V{X}<\infty
\eqv \samplemean \qmconv \mu$
\end{itemize}
\T{Slutzky's Theorem}
\begin{itemize}
\item $X_n \dconv X \text{ \T{and} } Y_n \pconv c
\imp X_n + Y_n \dconv X + c$
\item $X_n \dconv X \text{ \T{and} } Y_n \pconv c
\imp X_nY_n \dconv cX$
\item \T{In general}: $X_n \dconv X \text{ \T{and} } Y_n \dconv Y
\nimp X_n + Y_n \dconv X + Y$
\end{itemize}
\subsection{\T{Law of Large Numbers}}
\T{LLN Introduction}\\
\T{WLLN}
\[\samplemean \pconv \mu \qquad n\to\infty\]
\T{SLLN}
\[\samplemean \asconv \mu \qquad n\to\infty\]
\subsection{\T{Central Limit Theorem}}
\T{CLT Introduction}\\
\[ Z_n
:= \displaystyle\frac{\samplemean-\mu}{\sqrt{\V{\samplemean}}}
= \displaystyle\frac{\sqrt{n}(\samplemean - \mu)}{\sigma}
\dconv Z \qquad \text{\T{where} } Z\dist \norm[0,1]\]
\[\lim_{n\to\infty} \Pr{Z_n \le z} = \Phi(z) \qquad z \in \mathbb R\]
\T{CLT notations}
\begin{align*}
Z_n &\approx \norm[0,1] \\
\samplemean &\approx \norm[\mu,\frac{\sigma^2}{n}] \\
\samplemean - \mu &\approx \norm[0,\frac{\sigma^2}{n}] \\
\sqrt{n}(\samplemean - \mu) &\approx \norm[0,\sigma^2] \\
\frac{\sqrt{n}(\samplemean - \mu)}{n} &\approx \norm[0,1] \\
\end{align*}
\T{Continuity correction}
\[\Pr{\samplemean \le x} \approx
\Phi\left(\displaystyle\frac{x+\frac{1}{2}-\mu}{\sigma/\sqrt{n}}\right)\]
\[\Pr{\samplemean \ge x} \approx
1-\Phi\left(\displaystyle\frac{x-\frac{1}{2}-\mu}{\sigma/\sqrt{n}}\right)\]
\T{Delta method}
\[Y_n \approx \norm[\mu,\frac{\sigma^2}{n}] \imp
\transform(Y_n) \approx
\norm[\transform(\mu),
\left(\transform'(\mu)\right)^2\frac{\sigma^2}{n}]\]
\section{\T{Statistical Inference}}
\T{Statistical Inference Introduction}
\subsection{\T{Point Estimation}}
\begin{itemize}
\item \T{Point estimator}: $\that_n = g(X_1,\dots,X_n)$
\item $\bias(\that_n) = \E{\that_n}-\theta$
\item \T{Consistency}: $\that_n \pconv \theta$
\item \T{Sampling distribution}: $F(\that_n)$
\item \T{Standard error}: $\se(\that_n) = \sqrt{\V{\that_n}}$
\item \T{Mean squared error}: $\mse = \E{(\that_n-\theta)^2}
= \bias(\that_n)^2 + \V{\that_n}$
\item $\limn \bias(\that_n) = 0 \wedge \limn\se(\that_n) = 0
\imp \that_n$ \T{is consistent}
\item \T{Asymptotic normality}:
$\displaystyle\frac{\that_n-\theta}{\se} \dconv \norm[0,1]$
\item \T{Slutzky Note}
\end{itemize}
\subsection{\T{Normal-Based Confidence Interval}}
\T{Suppose} $\that_n \approx \norm[\theta,\sehat^2]$.
\T{Let} $\zat = \Phi^{-1}(1-(\alpha/2))$,
i.e., $\Pr{Z > \zat} = \alpha/2$ \T{and} $\Pr{-\zat < Z < \zat} = 1-\alpha$
\T{where} $Z\dist\norm[0,1]$.
\T{Then} \[C_n = \that_n \pm \zat\sehat\]
\subsection{\T{Empirical distribution}}
\T{Empirical Distribution Function}
\[\Fnhat(x) = \displaystyle\frac{\sumin I(X_i \le x)}{n}\]
\[I(X_i \le x) = \begin{cases}
1 & X_i \le x \\
0 & X_i > x
\end{cases}\]
\T{Properties} (\T{for any fixed x})
\begin{itemize}
\item $\E{\Fnhat} = F(x)$
\item $\V{\Fnhat} = \displaystyle\frac{F(x)(1-F(x))}{n}$
\item $\mse = \displaystyle\frac{F(x)(1-F(x))}{n} \dconv 0$
\item $\Fnhat \pconv F(x)$
\end{itemize}
\T{Dvoretzky-Kiefer-Wolfowitz inequality} ($X_1,\dots,X_n\dist F$)
\[\Pr{\sup_x\left|F(x)-\Fnhat(x)\right| > \varepsilon} =
2e^{-2n\varepsilon^2}\]
\T{Nonparametric confidence band}
\begin{align*}
L(x) &= \max\{\Fnhat-\epsilon_n, 0\} \\
U(x) &= \min\{\Fnhat+\epsilon_n, 1\} \\
\epsilon &=
\sqrt{\displaystyle\frac{1}{2n}\log\left( \frac{2}{\alpha} \right)} \\
\end{align*}
\[\Pr{L(x) \le F(x) \le U(x) \;\forall x} \ge 1-\alpha\]
\subsection{\T{Statistical Functionals}}
\begin{itemize}
\item \T{Statistical functional}: $T(F)$
\item \T{Plug-in estimator of theta}: $\that_n = T(\Fnhat)$
\item \T{Linear functional}: $T(F) = \int \transform(x)\dfx$
\item \T{Plug-in estimator for linear functional}: \\
\[T(\Fnhat)
= \displaystyle\int \transform(x)\dfhatx
= \frac{1}{n}\sumin \transform(X_i)\]
\item \T{Often}: $T(\Fnhat) \approx \norm[T(F),\sehat^2]$ \imp
$T(\Fnhat) \pm \zat\sehat$
\item $p^\mathrm{th}$ \T{quantile}: $F^{-1}(p) = \inf\{x:F(x) \ge p\}$
\item $\mhat = \samplemean$
\item $\shat^2 = \displaystyle\frac{1}{n-1}\sumin
(X_i-\samplemean)^2$
\item $\khat =
\displaystyle\frac{\frac{1}{n}\sumin(X_i-\mhat)^3}{\shat^3j}$
\item $\rhohat = \displaystyle\frac{\sumin(X_i-\samplemean)(Y_i-\bar{Y}_n)}%
{\sqrt{\sumin(X_i-\samplemean)^2}\sqrt{\sumin(Y_i-\bar{Y}_n)}}$
\end{itemize}
\section{\T{Parametric Inference}}
\T{Parametric Inference Introduction}
\subsection{\T{Method of Moments}}
\T{jth moment}
\[\alpha_j(\theta) = \E{X^j} = \displaystyle\int x^j \dfx\]
\T{jth sample moment}
\[\ahat_j = \displaystyle\frac{1}{n}\sumin X_i^j\]
\T{Method of Moments estimator}
\begin{align*}
\alpha_1(\theta) &= \ahat_1 \\
\alpha_2(\theta) &= \ahat_2 \\
\vdots &= \vdots \\
\alpha_k(\theta) &= \ahat_k
\end{align*}
\begin{titemize}{\T{Properties of the MoM estimator}}
\item $\that_n$ \T{exists with probability tending to 1}
\item \T{Consistency}: $\that_n \pconv \theta$
\item \T{Asymptotic normality}:
\[\sqrt{n}(\that-\theta) \dconv \norm[0,\Sigma]\]
\T{where} $\Sigma = g\E{YY^T}g^T$, $Y = (X,X^2,\dots,X^k)^T$,\\
$g = (g_1,\dots,g_k)$ \T{and}
$g_j = \frac{\partial}{\partial\theta}\alpha_j^{-1}(\theta)$
\end{titemize}
\subsection{\T{Maximum Likelihood}}
\T{Likelihood}: $\Lln : \Theta \to [0,\infty)$
\[\Lln(\theta) = \displaystyle\prodin f(X_i;\theta)\] \\
\T{Log-likelihood}
\[\lln(\theta) = \log \Lln(\theta) = \sumin \log f(X_i;\theta)\]
\T{Maximum likelihood estimator}
\[\Lln(\that_n) = \sup_\theta \Lln(\theta)\]
\T{Score function}
\[s(X;\theta) = \frac{\partial}{\partial\theta}\log f(X;\theta)\]
\T{Fisher information}
\[I(\theta) = \V[\theta]{s(X;\theta)}\]
\[I_n(\theta) = nI(\theta)\]
\T{Fisher information} (\T{exponential family})
\[I(\theta) = \E[\theta]{-\frac{\partial}{\partial\theta} s(X;\theta)}\]
\T{Observed Fisher information}
\[I_n^{obs}(\theta)
= -\frac{\partial^2}{\partial\theta^2} \sumin\log f(X_i;\theta)\]
\T{Properties of the MLE}
\begin{itemize}
\item \T{Consistency}: $\that_n \pconv \theta$
\item \T{Equivariance}:
$\that_n$ \T{is the} \mle
\imp $\transform(\that_n)$ \T{ist the} \mle \T{of} $\transform(\theta)$
\item \T{Asymptotic normality}:
\begin{enumerate}
\item $\se \approx \sqrt{1/I_n(\theta)}$
\[\frac{(\that_n - \theta)}{\se} \dconv \norm[0,1]\]
\item $\sehat \approx \sqrt{1/I_n(\that_n)}$
\[\frac{(\that_n - \theta)}{\sehat} \dconv \norm[0,1]\]
\end{enumerate}
\item \T{Asymptotic optimality}
\[\are(\ttil_n, \that_n)
= \frac{\V{\that_n}}{\V{\ttil_n}}
\le 1\]
\item \T{Approximately the Bayes estimator}
\end{itemize}
\subsubsection{\T{Delta Method}}
\T{If} $\tau=\transform(\that)$
\T{where} \transform \T{is differentiable}
\T{and} $\transform'(\theta) \neq 0$:
\[\frac{(\widehat{\tau}_n-\tau)}{\sehat(\widehat{\tau})} \dconv \norm[0,1]\]
\T{where} $\widehat{\tau} = \transform(\that)$
\T{is the} \mle \T{of} $\tau$ \T{and}
\[\sehat = \left|\transform'(\that)\right|\sehat(\that_n)\]
\subsection{\T{Multiparameter Models}}
\T{Let} $\theta=(\theta_1,\dots,\theta_k)$
\T{and} $\that= (\that_1,\dots,\that_k)$ \T{be the} \mle.
\[H_{jj} = \frac{\partial^2 \lln}{\partial\theta^2} \qquad
H_{jk} = \frac{\partial^2 \lln}{\partial\theta_j\partial\theta_k} \qquad\]
\T{Fisher information matrix}
\[I_n(\theta) = -\begin{bmatrix}
\E[\theta]{H_{11}} & \cdots & \E[\theta]{H_{1k}} \\
\vdots & \ddots & \vdots \\
\E[\theta]{H_{k1}} & \cdots & \E[\theta]{H_{kk}}
\end{bmatrix}\]
\T{Under appropriate regularity conditions}
\[(\that-\theta) \approx \norm[0,J_n]\]
\T{with} $J_n(\theta) = I_n^{-1}$.
\T{Note on component convergence}
\subsubsection{\T{Multiparameter delta method}}
\T{Let} $\tau = \transform(\theta_1,\dots,\theta_k)$
\T{and} \T{let the gradient of transform be}
\[\nabla\transform = \begin{pmatrix}
\displaystyle\frac{\partial\transform}{\partial\theta_1} \\
\vdots \\
\displaystyle\frac{\partial\transform}{\partial\theta_k} \\
\end{pmatrix}\]
\T{Suppose} $\bigl.\nabla\transform\bigr|_{\theta=\that} \neq 0$ \T{and}
$\widehat{\tau} = \transform(\that)$. \T{Then},
\[\frac{(\widehat{\tau}-\tau)}{\sehat(\widehat{\tau})} \dconv \norm[0,1]\]
\T{where}
\[\sehat(\widehat{\tau}) = \sqrt{\left( \widehat{\nabla}\transform \right)^T
\widehat{J}_n\left( \widehat{\nabla}\transform \right)}\]
\T{and} $\widehat{J}_n = J_n(\that)$ \T{and} $\widehat{\nabla}\transform =
\bigl.\nabla\transform\bigr|_{\theta=\that}$.
\subsection{\T{Parametric Bootstrap}}
\T{Parametric Bootstrap Note}