probcs.tex

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% filename: probcs.tex                 %
% version: 1.01                        %
% date: 6th November 2010              %
% author: Peleg Michaeli               %
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\documentclass[10pt,landscape]{article}
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\begin{document}

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\footnotesize
\begin{multicols}{4}
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\setlength{\premulticols}{1pt}
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\newcommand{\p}[1]{\mathbb{P}\left({#1}\right)}
\newcommand{\var}[1]{\text{Var}\left({#1}\right)}
\newcommand{\e}[1]{\mathbb{E}\left({#1}\right)}
\newcommand{\cov}[1]{\text{Cov}\left({#1}\right)}

\begin{center}
     \Large{\textbf{Probability Cheat Sheet}} \\
\end{center}

\section{Distributions}
\subsection{Unifrom Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{U\left[a,b\right]}$ \\
\verb!cdf!          & $\displaystyle{\frac{x-a}{b-a}}$ for $\displaystyle{x\in\left[a,b\right]}$ \\
\verb!pdf!          & $\displaystyle{\frac{1}{b-a}}$ for $\displaystyle{x\in\left[a,b\right]}$ \\
\verb!expectation!  & $\displaystyle{\frac{1}{2}\left(a+b\right)}$ \\
\verb!variance!     & $\displaystyle{\frac{1}{12}\left(b-a\right)^2}$ \\
\verb!mgf!          & $\displaystyle{\frac{e^{tb}-e^{ta}}{t\left(b-a\right)}}$
\end{tabular}

\verb!story:! all intervals of the same length on the distribution's
support are equally probable.


\subsection{Gamma Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{Gamma\left(k,\theta\right)}$ \\
\verb!pdf!          & $\displaystyle{\frac{\theta^k x^{k-1}e^{-\theta
                       x}}{\Gamma\left(k\right)}\mathbb{I}_{x>0}}$ \\
                    & $\displaystyle{\Gamma\left(k\right)=\int_0^\infty
                       x^{k-1}e^{-x}dx}$ \\
\verb!expectation!  & $\displaystyle{k\theta}$ \\
\verb!variance!     & $\displaystyle{k\theta^2}$ \\
\verb!mgf!          & $\displaystyle{\left(1-\theta t\right)^{-k}}$ for
                      $\displaystyle{t<\frac{1}{\theta}}$ \\
\verb!ind. sum!     & $\displaystyle{\sum_{i=1}^nX_i\sim Gamma\left(\sum_{i=1}^nk_i,\theta\right)}$
\end{tabular}

\verb!story:! the sum of k independent exponentially distributed random variables, each of which has
a mean of $\theta$ (which is equivalent to a rate parameter of $\theta^{-1}$).


\subsection{Geometric Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{G\left(p\right)}$ \\
\verb!cdf!          & $\displaystyle{1-\left(1-p\right)^k}$ for $\displaystyle{k\in\mathbb{N}}$ \\
\verb!pmf!          & $\displaystyle{\left(1-p\right)^{k-1}p}$ for $\displaystyle{k\in\mathbb{N}}$ \\
\verb!expectation!  & $\displaystyle{\frac{1}{p}}$ \\
\verb!variance!     & $\displaystyle{\frac{1-p}{p^2}}$ \\
\verb!mgf!          & $\displaystyle{\frac{pe^t}{1-\left(1-p\right)e^t}}$
\end{tabular}

\verb!story:! the number X of Bernoulli trials needed to get one success. Memoryless.


\subsection{Poisson Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{Poisson\left(\lambda\right)}$ \\
\verb!cdf!          & $\displaystyle{e^{-\lambda}\sum_{i=0}^k\frac{\lambda^i}{i!}}$ \\
\verb!pmf!          & $\displaystyle{\frac{\lambda^k}{k!}\cdot e^{-\lambda}}$ for $\displaystyle{k\in\mathbb{N}}$ \\
\verb!expectation!  & $\displaystyle{\lambda}$ \\
\verb!variance!     & $\displaystyle{\lambda}$ \\
\verb!mgf!          & $\displaystyle{\exp{\left(\lambda\left(e^t-1\right)\right)}}$ \\
\verb!ind. sum!     & $\displaystyle{\sum_{i=1}^nX_i\sim Poisson\left(\sum_{i=1}^n\lambda_i\right)}$
\end{tabular}

\verb!story:! the probability of a number of events occurring in a fixed period of time if these
events occur with a known average rate and independently of the time since the last event.


\subsection{Normal Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{N\left(\mu,\sigma^2\right)}$ \\
\verb!pdf!          & $\displaystyle{\frac{1}{\sqrt{2\pi\sigma^2}}
                       e^{-\left(x-\mu\right)^2/\left(2\sigma^2\right)}}$ \\
\verb!expectation!  & $\displaystyle{\mu}$ \\
\verb!variance!     & $\displaystyle{\sigma^2}$ \\
\verb!mgf!          & $\displaystyle{\exp{\left(\mu t+\frac{1}{2}\sigma^2t^2\right)}}$ \\
\verb!ind. sum!     & $\displaystyle{\sum_{i=1}^nX_i\sim
                       N\left(\sum_{i=1}^n\mu_i,\sum_{i=1}^n\sigma_i^2\right)}$
\end{tabular}

\verb!story:! describes data that cluster around the mean.


\subsection{Standard Normal Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{N\left(0,1\right)}$ \\
\verb!cdf!          & $\displaystyle{\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}dt}$ \\
\verb!pdf!          & $\displaystyle{\frac{1}{\sqrt{2\pi}}e^{-x^2/2}}$ \\
\verb!expectation!  & $\displaystyle{\frac{1}{\lambda}}$ \\
\verb!variance!     & $\displaystyle{\frac{1}{\lambda^2}}$ \\
\verb!mgf!          & $\displaystyle{\exp{\left(\frac{t^2}{2}\right)}}$
\end{tabular}

\verb!story:! normal distribution with $\mu=0$ and $\sigma=1$.

\subsection{Exponential Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{exp\left(\lambda\right)}$ \\
\verb!cdf!          & $\displaystyle{1-e^{-\lambda x}}$ for $\displaystyle{x\ge 0}$ \\
\verb!pdf!          & $\displaystyle{\lambda e^{-\lambda x}}$ for $\displaystyle{x\ge 0}$ \\
\verb!expectation!  & $\displaystyle{\frac{1}{\lambda}}$ \\
\verb!variance!     & $\displaystyle{\frac{1}{\lambda^2}}$ \\
\verb!mgf!          & $\displaystyle{\frac{\lambda}{\lambda-t}}$ \\
\verb!ind. sum!     & $\displaystyle{\sum_{i=1}^kX_i\sim Gamma\left(k,\lambda\right)}$ \\
\verb!minimum!      & $\displaystyle{\sim exp\left(\sum_{i=1}^k\lambda_i\right)}$
\end{tabular}

\verb!story:! the amount of time until some specific event occurs, starting from now,
being memoryless.


\subsection{Binomial Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation!     & $\displaystyle{Bin(n,p)}$ \\
\verb!cdf!          & $\displaystyle{\sum_{i=0}^k {n \choose i} p^i\left(1-p\right)^{n-i}}$ \\
\verb!pmf!          & $\displaystyle{{n \choose i} p^i\left(1-p\right)^{n-i}}$ \\
\verb!expectation!  & $\displaystyle{np}$ \\
\verb!variance!     & $\displaystyle{np\left(1-p\right)}$ \\
\verb!mgf!          & $\displaystyle{\left(1-p+pe^t\right)^n}$
\end{tabular}

\verb!story:! the discrete probability distribution of the number of successes in a sequence of $n$
independent yes/no experiments, each of which yields success with probability $p$.


\rule{\linewidth}{2.5pt}

\section{Basics}
\subsection{Comulative Distribution Function}
$\displaystyle{F_X\left(x\right)=\p{X\le x}}$

\subsection{Probability Density Function}
$\displaystyle{F_X\left(x\right)=\int_{-\infty}^{\infty}f_X\left(t\right)dt}$

$\displaystyle{\int_{-\infty}^{\infty}f_X\left(t\right)dt=1}$

$\displaystyle{f_X\left(x\right)=\frac{d}{dx}F_X\left(x\right)}$

\subsection{Quantile Function}
The function $X^*:\left[0,1\right]\to\mathbb{R}$ for which for any $p\in\left[0,1\right]$,
$F_X\left(X^*\left(p\right)^{-}\right)\le p \le F_X\left(X^*\left(p\right)\right)$

$\displaystyle{F_{X^*}=F_X}$

$\displaystyle{\e{X^*}=\e{X}}$

\subsection{Expectation}
$\displaystyle{\e{X}=\int_0^1X^*(p)dp}$

$\displaystyle{\e{X}=\int_{-\infty}^0F_X\left(t\right)dt+\int_0^{\infty}\left(1-F_X\left(t\right)\right)dt}$

$\displaystyle{\e{X}=\int_{-\infty}^{\infty}xf_X{x}dx}$

$\displaystyle{\e{g\left(X\right)}=\int_{-\infty}^{\infty}g\left(x\right)f_X{x}dx}$

$\displaystyle{\e{aX+b}=a\e{X}+b}$

\subsection{Variance}
$\displaystyle{\var{X}=\e{X^2}-\left(\e{X}\right)^2}$

$\displaystyle{\var{X}=\e{\left(X-\e{X}\right)^2}}$

$\displaystyle{\var{aX+b}=a^2\var{X}}$

\subsection{Standard Deviation}
$\displaystyle{\sigma\left(X\right)=\sqrt{\var{X}}}$

\subsection{Covariance}
$\displaystyle{\cov{X,Y}=\e{XY}-\e{X}\e{Y}}$

$\displaystyle{\cov{X,Y}=\e{\left(X-\e{x}\right)\left(Y-\e{Y}\right)}}$

$\displaystyle{\var{X+Y}=\var{X}+\var{Y}+2\cov{X,Y}}$

\subsection{Correlation Coefficient}
$\displaystyle{\rho_{X,Y}=\frac{\cov{X,Y}}{\sigma_X,\sigma_Y}}$

\subsection{Moment Generating Function}
$\displaystyle{M_X\left(t\right)=\e{e^{tX}}}$

$\displaystyle{\e{X^n}=M_X^{\left(n\right)}\left(0\right)}$

$\displaystyle{M_{aX+b}\left(t\right)=e^{tb}M_{aX}\left(t\right)}$


\rule{\linewidth}{2.5pt}

\pagebreak
\section{Joint Distribution}
$\displaystyle{\mathbb{P}_{X,Y}\left(B\right)=\mathbb{P}\left(\left(X,Y\right)\in B\right)}$

$\displaystyle{F_{X,Y}\left(x,y\right)=\mathbb{P}\left(X\le x,Y\le y\right)}$


\subsection{Joint Density}
$\displaystyle{\mathbb{P}_{X,Y}\left(B\right)=\iint_B f_{X,Y}\left(s,t\right)dsdt}$

$\displaystyle{F_{X,Y}\left(x,y\right)=\int_{-\infty}^x\int_{-\infty}^yf_{X,Y}\left(s,t\right)dtds}$

$\displaystyle{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{X,Y}\left(s,t\right)dsdt=1}$

\subsection{Marginal Distributions}
$\displaystyle{\mathbb{P}_X\left(B\right) = \mathbb{P}_{X,Y}\left(B\times\mathbb{R}\right)}$

$\displaystyle{\mathbb{P}_Y\left(B\right) = \mathbb{P}_{X,Y}\left(\mathbb{R}\times Y\right)}$

$\displaystyle{F_X\left(a\right) =
\int_{-\infty}^a\int_{-\infty}^{\infty}f_{X,Y}\left(s,t\right)dtds}$

$\displaystyle{F_Y\left(b\right) =
\int_{-\infty}^b\int_{-\infty}^{\infty}f_{X,Y}\left(s,t\right)dsdt}$

\subsection{Marginal Densities}
$\displaystyle{f_X\left(s\right)=\int_{-\infty}^{\infty}f_{X,Y}(s,t)dt}$

$\displaystyle{f_Y\left(t\right)=\int_{-\infty}^{\infty}f_{X,Y}(s,t)ds}$

\subsection{Joint Expectation}
$\displaystyle{\mathbb{E}\left(\varphi\left(X,Y\right)\right)=\iint_{\mathbb{R}^2}\varphi\left(x,y\right)f_{X,Y}\left(x,y\right)dxdy}$

\rule{\linewidth}{2.5pt}


\section{Independent r.v.}

$\displaystyle{\mathbb{P}\left(X\le x,Y\le y\right)=\mathbb{P}\left(X\le
 x\right)\mathbb{P}\left(Y\le y\right)}$

$\displaystyle{F_{X,Y}\left(x,y\right)=F_X\left(x\right)F_Y\left(y\right)}$

$\displaystyle{f_{X,Y}\left(s,t\right)=f_X\left(s\right)f_Y\left(t\right)}$

$\displaystyle{\e{XY}=\e{X}\e{Y}}$

$\displaystyle{\var{X+Y}=\var{X}+\var{Y}}$

Independent events:

$\displaystyle{\p{A\cap B}=\p{A}\p{B}}$

\rule{\linewidth}{2.5pt}

\section{Conditional Probability}
$\displaystyle{\mathbb{P}\left(A\mid B\right)=\frac{\mathbb{P}\left(A\cap
B\right)}{\mathbb{P}\left(B\right)}}$

\verb!bayes! $\displaystyle{\p{A\mid B}=\frac{\p{B\mid A}\p{A}}{\p{B}}}$

\subsection{Conditional Density}
$\displaystyle{f_{X\mid Y=y}\left(x\right)=\frac{f_{X,Y}\left(x,y\right)}{f_Y\left(y\right)}}$

$\displaystyle{f_{X\mid Y=n}\left(x\right)=\frac{f_X\left(x\right)\mathbb{P}\left(Y=n\mid
 X=x\right)}{\mathbb{P}\left(Y=n\right)}}$

$\displaystyle{F_{X\mid Y=y} = \int_{-\infty}^{x}f_{X\mid Y=y}\left(t\right)dt}$

\subsection{Conditional Expectation}
$\displaystyle{\e{X\mid Y=y}=\int_{-\infty}^\infty xf_{X\mid Y=y}\left(x\right)dx}$

$\displaystyle{\e{\e{X\mid Y}} = \e{X}}$

$\displaystyle{\p{Y=n}=\e{\mathbb{I}_{Y=n}}=\e{\e{\mathbb{I}_{Y=n}\mid X}}}$

\rule{\linewidth}{2.5pt}


\section{Sequences and Limits}
$\displaystyle{\limsup A_n = \left\{A_n \text{ i.o.}\right\} = \bigcap_{m=1}^\infty
 \bigcup_{n=m}^\infty A_n}$

$\displaystyle{\liminf A_n = \left\{A_n \text{ eventually}\right\} = \bigcup_{m=1}^\infty
 \bigcap_{n=m}^\infty A_n}$

$\displaystyle{\liminf A_n \subseteq \limsup A_n}$

$\displaystyle{\left(\limsup A_n\right)^c = \liminf A_n^c}$

$\displaystyle{\left(\liminf A_n\right)^c = \limsup A_n^c}$

$\displaystyle{\p{\limsup A_n} = \lim_{n\to\infty}\p{\bigcup_{n=m}^\infty A_n}}$

$\displaystyle{\p{\liminf A_n} = \lim_{n\to\infty}\p{\bigcap_{n=m}^\infty A_n}}$

\subsection{Borel-Cantelli Lemma}
$\displaystyle{\sum_{n=1}^\infty\p{A_n}<\infty \Rightarrow \p{\limsup A_n}=0}$

And if $A_n$ are independent:

$\displaystyle{\sum_{n=1}^\infty\p{A_n}=\infty \Rightarrow \p{\limsup A_n}=1}$


\rule{\linewidth}{2.5pt}


\section{Convergence}
\subsection{Convergence in Probability}
\begin{tabular}{@{}ll@{}}
\verb!notation! & $\displaystyle{X_n\xrightarrow{p}X}$ \\
\verb!meaning!  & $\displaystyle{\lim_{n\to\infty}\p{\left|X_n-X\right|>\varepsilon}=0}$
\end{tabular}

\subsection{Convergence in Distribution}
\begin{tabular}{@{}ll@{}}
\verb!notation! & $\displaystyle{X_n\xrightarrow{D}X}$ \\
\verb!meaning!  & $\displaystyle{\lim_{n\to\infty}F_n\left(x\right)=F\left(x\right)}$
\end{tabular}

\subsection{Almost Sure Convergence}
\begin{tabular}{@{}ll@{}}
\verb!notation! & $\displaystyle{X_n\xrightarrow{a.s.}X}$ \\
\verb!meaning!  & $\displaystyle{\p{\lim_{n\to\infty}X_n=X}=1}$
\end{tabular}

\subsubsection{Criteria for a.s. Convergence}
\begin{itemize}\setlength{\itemindent}{-1.5em}
  \item $\displaystyle{\forall\varepsilon \exists N \forall n>N:\p{\left|X_n-X\right|<\varepsilon}>1-\varepsilon}$
  \item $\displaystyle{\forall\varepsilon \p{\limsup\left(\left|X_n-X\right|>\varepsilon\right)}=0}$
  \item $\displaystyle{\forall\varepsilon
  \sum_{n=1}^\infty\p{\left|X_n-X\right|>\varepsilon}<\infty}$ (by B.C.)
\end{itemize}

\subsection{Convergence in $L_p$}
\begin{tabular}{@{}ll@{}}
\verb!notation! & $\displaystyle{X_n\xrightarrow{L_p}X}$ \\
\verb!meaning!  & $\displaystyle{\lim_{n\to\infty}\e{\left|X_n-X\right|^p}=0}$
\end{tabular}

\subsection{Relationships}
$$\begin{matrix}
  \xrightarrow{L_q}  & \underset{q>p\ge 1}{\Rightarrow} &  \xrightarrow{L_p}  &             & \\
                     &                                  &     \Downarrow      &             & \\
  \xrightarrow{a.s.} &            \Rightarrow           & \xrightarrow{\ p\ } & \Rightarrow & \xrightarrow{\ D\ }
  \end{matrix}$$
If $X_n\xrightarrow{D} c$ then $X_n\xrightarrow{p} c$

If $\displaystyle{X_n\xrightarrow{p}X}$ then there exists a subsequence $n_k$ s.t.
  $\displaystyle{X_{n_k}\xrightarrow{a.s.}X}$

\subsection{Laws of Large Numbers}
If $X_i$ are i.i.d. r.v.,
\begin{tabular}{@{}ll@{}}
\verb!weak law!   & $\displaystyle{\overline{X_n}\xrightarrow{p}\e{X_1}}$ \\
\verb!strong law! & $\displaystyle{\overline{X_n}\xrightarrow{a.s.}\e{X_1}}$
\end{tabular}

\subsection{Central Limit Theorem}
$\displaystyle{\frac{S_n-n\mu}{\sigma\sqrt{n}}\xrightarrow{D}N\left(0,1\right)}$

If $t_n\to t$, then

$\displaystyle{\p{\frac{S_n-n\mu}{\sigma\sqrt{n}}\le t_n}\to\Phi\left(t\right)}$


\rule{\linewidth}{2.5pt}


\section{Inequalities}
\subsection{Markov's inequality}
$\displaystyle{\p{\left|X\right|\ge t}\le\frac{\e{\left|X\right|}}{t}}$

\subsection{Chebyshev's inequality}
$\displaystyle{\p{\left|X-\e{X}\right|\ge\varepsilon}\le\frac{\var{X}}{\varepsilon^2}}$

\subsection{Chernoff's inequality}
Let $X\sim Bin(n,p)$; then:
$\displaystyle{\p{X-\e{X}>t\sigma\left(X\right)}<e^{-t^2/2}}$

Simpler result; for every $X$:
$\displaystyle{\p{X\ge a} \le M_X\left(t\right)e^{-ta}}$

\subsection{Jensen's inequality}
for $\varphi$ a convex function,
$\displaystyle{\varphi\left(\e{X}\right)\le\e{\varphi\left(X\right)}}$


\rule{\linewidth}{2.5pt}


\section{Miscellaneous}
$\displaystyle{\mathbb{E}\left(Y\right)<\infty \iff
\sum_{n=0}^\infty\mathbb{P}\left(Y>n\right)<\infty}$ ($Y\ge 0$)

$\displaystyle{\mathbb{E}\left(X\right)=\sum_{n=0}^\infty\mathbb{P}\left(X>n\right)}$
($X\in\mathbb{N}$)

$\displaystyle{X\sim U\left(0,1\right) \iff -\ln{X}\sim exp\left(1\right)}$

\subsection{Convolution}
For ind. $X,Y$, $Z=X+Y$:
$\displaystyle{f_Z\left(z\right) = \int_{-\infty}^\infty f_X\left(s\right)f_Y\left(z-s\right)ds}$

\subsection{Kolmogorov's 0-1 Law}
If $A$ is in the tail $\sigma$-algebra $\mathcal{F}^t$, then $\p{A}=0$ or $\p{A}=1$

\subsection{Ugly Stuff}
cdf of Gamma distribution:

$\displaystyle{\int_0^t\frac{\theta^kx^{k-1}e^{-\theta k}}{\left(k-1\right)!}dx}$

\vspace{5em}
\rule{0.3\linewidth}{0.25pt}
\scriptsize

This cheatsheet was made by Peleg Michaeli in November 2010, using \LaTeX.

version: 1.01

comments: peleg.michaeli@math.tau.ac.il


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