% STA 260 Formula sheet

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\begin{document}

\begin{center}   
{\LARGE \hspace{20mm} \textbf{STA 260 Formulas and Tables}}\\
\end{center}
\vspace{7 mm}

\noindent
%\renewcommand{\arraystretch}{2.0}
\begin{tabular}{lll}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Math %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$ \displaystyle  \sum_{k=j}^{\infty} a^k = \frac{a^j}{1-a}$ &
$  \displaystyle \sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$ & 
$\displaystyle (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}$ \\
&&\\ % Space
$\Gamma(\alpha) = \int_0^\infty e^{-t} t^{\alpha-1} \, dt$
& $\Gamma(\alpha+1) = \alpha \, \Gamma(\alpha)$ &
$\Gamma(\frac{1}{2}) = \sqrt{\pi} $ \hspace{5mm} 
$\displaystyle \lim_{n \rightarrow \infty}\left(1 + \frac{x}{n}\right)^n = e^x$ \\
&&\\ % Space
$I_{_A}(x) \stackrel{def}{=} \left\{ \begin{array}{ll} 
   1 & \mbox{for } x \in A  \\
   0 & \mbox{for } x \notin A   
   \end{array}  \right. $ &
$I(x \in A) \stackrel{def}{=} \left\{ \begin{array}{ll} 
   1 & \mbox{for } x \in A  \\
   0 & \mbox{for } x \notin A   
   \end{array}  \right. $   &
$g(x)I(x \in A) = \left\{ \begin{array}{ll} 
   g(x) & \mbox{for } x \in A  \\
   0 & \mbox{for } x \notin A   
   \end{array}  \right. $                       \\
&&\\ % Space
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Densities, PMFs and CDFs %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$F_{_X}(x) \stackrel{def}{=} P(X \leq x)$ & $F_{_X}(x) = \displaystyle \int_{-\infty}^x f_{_X}(t) \, dt$ &
$F_{_X}(x) = \displaystyle \sum_t I(t \leq x) \, p_{_X}(t)$ \\ 
&&\\ % Space
$p_{_X}(x) = \displaystyle \sum_y p_{_{X,Y}}(x,y)$ &
$p_{_{Y|X}}(y|x) \stackrel{def}{=} \frac{p_{_{X,Y}}(x,y)}{p_{_X}(x)}$ & 
$p_{_X}(x) = \displaystyle \sum_y p_{_{X|Y}}(x|y) \, p_{_Y}(y)$  \\
&&\\ % Space
$f_{_X}(x) = \displaystyle \int_{-\infty}^\infty f_{_{X,Y}}(x,y) \, dy$ &
$f_{_{Y|X}}(y|x) \stackrel{def}{=} \frac{f_{_{X,Y}}(x,y)} {f_{_X}(x)}$ &
$f_{_X}(x) = \displaystyle \int_{-\infty}^\infty f_{_{X|Y}}(x|y) \,
f_{_Y}(y) \, dy$   \\
&&\\ % Space
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Expected value, variance and covariance %%%%%%%%%%%%%%%%%
$E(X) \stackrel{def}{=} \displaystyle \sum_x x \, p_{_X}(x)$ &
$E(g(X)) = \displaystyle \sum_x g(x) \, p_{_X}(x)$  & 
$E(g(X,Y)) = \displaystyle \sum_x \sum_y g(x,y) \, p_{_{X,Y}}(x,y)$   \\
&&\\ % Space
$E(X) \stackrel{def}{=} \displaystyle \int_{-\infty}^\infty x \, f_{_X}(x) \, dx$ &
$E(g(X)) = \displaystyle \int_{-\infty}^\infty g(x) \, f_{_X}(x) \, dx$ & 
$E(g(X,Y)) = \displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty g(x,y) \, 
             f_{_{X,Y}}(x,y) \, dx \, dy$ \\
&&\\ % Space
$Var(X) \stackrel{def}{=} E\left( (X-\mu)^2 \right)$ &
$Var(X) = E(X^2)-[E(X)]^2$ & $Var(aX) = a^2Var(X)$ \\
&&\\ % Space
\multicolumn{3}{l} {$Var(aX+bY) = a^2Var(X)+b^2Var(Y)+2abCov(X,Y)$} \\
&&\\ % Space
\multicolumn{3}{l} {$Cov(X,Y) \stackrel{def}{=} E[(X-\mu_{_X})(Y-\mu_{_Y})]
\hspace{10mm} Cov(X,Y) = E(XY) - E(X)E(Y)$  } \\
&&\\ % Space
\multicolumn{3}{l} {$Cov(aX,bY) = ab \, Cov(X,Y)$  \hspace{10mm}
$Cov\left( \sum_{i=1}^n X_i, \sum_{j=1}^m Y_j  \right) = 
\sum_{i=1}^n \sum_{j=1}^m Cov(X_i,Y_j)$} \\
&&\\ % Space
\multicolumn{3}{l} {$Var\left(\sum_{i=1}^n a_iX_i \right) = \sum_{i=1}^n a_i^2Var(X_i) 
\, + \, \sum\sum_{i \neq j} a_ib_j Cov(X_i,X_j)$ } \\
&&\\ % Space
\multicolumn{3}{l} {If $X_1, \ldots, X_n$ are independent, 
                    $Var\left(\sum_{i=1}^n X_i \right) = \sum_{i=1}^n Var(X_i)$} \\
&&\\ % Space
%%%%%%%%%%%%%%%%%%%%%%%%%%%% MGFs %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$M_{_X}(t) \stackrel{def}{=} E(e^{Xt})$ &  $M_{_X}^{(k)}(0) = E(X^k)$~ & \\
&&\\ % Space
$M_{_{aX}}(t) = M_{_X}(at)$ &  
\multicolumn{2}{l} {$M_{_{\sum X_i}}(t) = \prod_{i=1}^n M_{_{X_i}}(t)$ if the $X_i$ are independent.} \\
% &&\\ % Space
% \multicolumn{3}{l} { If  $Y=Y_1+Y_2$ with $Y_1$ and $Y_2$ independent, $Y\sim\chi^2(\nu_1+\nu_2)$, $Y_2\sim\chi^2(\nu_2)$ then $Y_1\sim\chi^2(\nu_1)$.}\\
\end{tabular}
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\pagebreak

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\begin{tabular}{lcl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Limits %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\multicolumn{3}{l} {\emph{Convergence in probability}: } \\
\multicolumn{3}{l} {$T_n \stackrel{p}{\rightarrow} c$ means for all $\epsilon>0$, $\displaystyle \lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0
\Leftrightarrow \lim_{n \rightarrow \infty}P\{|T_n-c| < \epsilon\} = 1$ } \\
\multicolumn{3}{l} {Variance rule: If $\displaystyle \lim_{n \rightarrow \infty}E(T_n) = c$ and $\displaystyle \lim_{n \rightarrow \infty}Var(T_n) = 0$, then $T_n \stackrel{p}{\rightarrow} c$. } \\
\multicolumn{3}{l} {Law of Large Numbers: $\overline{X}_n \stackrel{p}{\rightarrow} \mu = E(X_i)$. } \\
\multicolumn{3}{l} {Continuous mapping: If $T_n \stackrel{p}{\rightarrow} c$ and $g(x)$ is continuous at $x=c$, then $g(T_n) \stackrel{p}{\rightarrow} g(c)$ } \\
\multicolumn{3}{l} {\emph{Convergence in Distribution}: } \\
\multicolumn{3}{l} {$X_n \stackrel{d}{\rightarrow} X$ means $\displaystyle \lim_{n \rightarrow \infty}F_{_{X_n}}(x) = F_{_X}(x)$ at every point where $F_{_X}(x)$ is continuous.} \\
$\overline{X}_n = \frac{1}{n}\sum_{i=1}^nX_i$ &&
$S^2 = \frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X}_n)^2$ \\
\multicolumn{3}{l} {Central Limit Theorem: 
If $X_1, \ldots, X_n \stackrel{ind}{\sim} \, ?(\mu,\sigma^2)$, then  $Z_n = \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma} \stackrel{d}{\rightarrow} Z \sim$ Normal (0,1).} \\
\multicolumn{3}{l} { \hspace{28mm}
If $X_1, \ldots, X_n \stackrel{ind}{\sim} \, ?(\mu,\sigma^2)$ and $\widehat{\sigma}^2_n \stackrel{p}{\rightarrow} \sigma^2$, then 
 $\frac{\sqrt{n}(\overline{X}_n-\mu)}{\widehat{\sigma}_n} \stackrel{d}{\rightarrow} Z \sim$ Normal (0,1).} \\
 $\overline{X}_n \pm z_{1-\alpha/2} \, \frac{\widehat{\sigma}_n}{\sqrt{n}}$ & 
 $T = \frac{Z}{\sqrt{Y/\nu}} \sim t(\nu)$ & 
$\overline{X} \pm t_{1-\alpha/2} \, \frac{S}{\sqrt{n}}$ \\
\end{tabular}
\renewcommand{\arraystretch}{1.0}

\vspace{2mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Distributions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\( \begin{array}{lllcc}

\mbox{Distribution} & ~~~~~~f_{_X}(x|\theta) \mbox{ or } p_{_X}(x|\theta) & M_{_X}(t) & E(X) & Var(X)   \\ \hline \\
\mbox{Bernoulli}     & \theta^x (1-\theta)^{1-x} I(x = 0, 1)
                     & \theta e^t + 1-\theta  
                     & \theta       & \theta (1-\theta)  \\[2 mm]
\mbox{Binomial} & \binom{m}{x} \theta^x (1-\theta)^{m-x} I(x = 0, \ldots, m)
                      & (\theta e^t + 1-\theta)^m  
                      & m\theta     & m \theta (1-\theta)  \\[2 mm]
\mbox{Poisson}  & \frac{e^{-\lambda}\lambda^x}{x!} I(x = 0, 1, \ldots)
                      & e^{\lambda(e^t-1)} 
                      & \lambda     & \lambda   \\[2 mm] 

\mbox{Geometric}  & (1-\theta)^x \theta \, I(x = 0, 1, \ldots)
                      & \theta\left( 1 - (1-\theta)e^t \right)^{-1}   & \frac{1-\theta}{\theta} &  \frac{1-\theta}{\theta^2} \\[2 mm] 

\mbox{Exponential}  & \lambda e^{-\lambda x} \, I(x > 0)
                      & (1 - \frac{t}{\lambda})^{-1} 
                      & \frac{1}{\lambda}      & \frac{1}{\lambda^2}  \\[2 mm] 

\mbox{Gamma}  & \frac{\lambda^\alpha}{\Gamma(\alpha)} 
                 e^{-\lambda x} x^{\alpha - 1} \, I(x > 0)
                      & (1 - \frac{t}{\lambda})^{-\alpha} 
                      & \frac{\alpha}{\lambda}    & \frac{\alpha}{\lambda^2}    \\[2mm] 

\mbox{Chi-squared } (\chi^2)  & \frac{1}{2^{\nu/2}\Gamma(\nu/2)} 
                 e^{-x/2} x^{\nu/2 - 1}I(x > 0)
                      & (1-2t)^{-\nu/2} 
                      & \nu     & 2 \nu \\[2 mm] 

\mbox{Normal}  & \frac{1}{\sigma \sqrt{2\pi}} 
                  e^{-\frac{(x-\mu)^2}{2 \sigma^2}}
                      & e^{\mu t + \frac{1}{2}\sigma^2 t^2} 
                      & \mu     & \sigma^2  \\[2 mm] 

\mbox{Uniform}  & \frac{1}{R-L} I(L < x < R) 
                      & \frac{e^{R t}-e^{L t}}{t(R - L)} 
                      & \frac{L+R}{2}     
                      & \frac{(R - L)^2}{12}  \\[2 mm] 
\mbox{Beta}  & \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \, 
               x^{\alpha-1} (1-x)^{\beta-1} \, I(0<x<1)
                      &  \mbox{Useless}
                      & \frac{\alpha}{\alpha+\beta}     
                      & \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}  \\[2 mm] 
\hline \\
\end{array} \)

\vspace{3mm}

\noindent
If $X \sim $ Exponential ($\lambda$), $F_{_X}(x) = 1-e^{-\lambda x} I(x>0)$. \hspace{5mm}
If $X \sim $ N($\mu,\sigma^2$), $\frac{X-\mu}{\sigma} \sim$ N(0,1)

\vspace{2mm}
\noindent
A \emph{random sample} is a collection of independent and identically distributed random variables $X_1, \ldots, X_n$. Write $X_1, \ldots, X_n \stackrel{i.i.d}{\sim} P_\theta,~\theta \in \Omega$. $\theta$ is the \emph{parameter}, and $\Omega$ is the \emph{parameter space}.

\vspace{2mm}
\noindent
An estimator $T_n = T_n(X_1, \ldots, X_n)$ is said to be \emph{unbiased} for $\theta$ if $E(T_n)=\theta$ for all $\theta \in \Omega$. 

\vspace{2mm}
\noindent
An estimator $T_n = T_n(X_1, \ldots, X_n)$ is said to be \emph{consistent} for $\theta$ if $T_n \stackrel{p}{\rightarrow} \theta$ for all $\theta \in \Omega$. 

\vspace{2mm}
\noindent
Method of Moments: Substitute sample moments for population moments and put hats on.

\vspace{2mm}
\noindent
Least squares: Minimize $Q(\theta) =  \sum_{i=1}^n \left(Y_i-E_\theta(Y_i)\right)^2$

\pagebreak

% Minmum and maximum?

\noindent
If $X_1, \ldots, X_n$ is a random sample from an Exponential$(\lambda)$ distribution, % then \\
    \begin{itemize}
        \item $\overline{X}_n \sim$ Gamma$(n,n\lambda)$.
        \item $Y = 2n\lambda\overline{X}_n \sim \chi^2(2n)$.
    \end{itemize} \vspace{3mm}

\noindent
If  $W=W_1+W_2$ with $W_1$ and $W_2$ independent, $W\sim\chi^2(\nu_1+\nu_2)$, $W_2\sim\chi^2(\nu_2)$ then $W_1\sim\chi^2(\nu_1)$. \vspace{3mm}    

\noindent
$t = \frac{Z}{\sqrt{W/\nu}} \sim t(\nu)$
\hspace{20mm}
$F = \frac{W_1/\nu_1}{W_2/\nu_2} \sim F(\nu_1,\nu_2)$ 
\vspace{3mm}

\noindent
If $X_1, \ldots, X_n$ is a random sample from a Normal$(\mu,\sigma^2)$ distribution, then
    \begin{itemize}
        \item $\widehat{\mu} = \overline{X}_n$ and 
        $\widehat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n(X_i-\overline{X}_n)^2 = \left(\frac{n-1}{n} \right)S^2$.
        \item $\overline{X}_n$ and $S^2$ are independent.
        \item $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)$. 
        \item $t = \frac{\sqrt{n}(\overline{X}-\mu)}{S} \sim t(n-1)$
    \end{itemize} \vspace{3mm}
    
% Two-sample
\noindent
If $X_1, \ldots, X_{n_1} \stackrel{i.i.d.}{\sim}$ Normal$(\mu_1,\sigma_1^2)$ and 
$Y_1, \ldots, Y_{n_2} \stackrel{i.i.d.}{\sim}$ Normal$(\mu_2,\sigma_2^2)$ with $X_i$ independent of $Y_i$, then
    \begin{itemize}
        \item $F = \frac{S^2_1/\sigma^2_1}{S^2_2/\sigma^2_2} \sim F(n_1-1,n_2-1)$.
        \item $T = \frac{\overline{X}-\overline{Y} - (\mu_1-\mu_2)}
               {S_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \sim t(n_1+n_2-2)$
               provided $\sigma^2_1=\sigma^2_2$,
               where $S_p = \sqrt{\frac{(n_1-1)S^2_1 + (n_2-1)S^2_2 }{n_1+n_2-2}}$.
    \end{itemize} \vspace{3mm}

\noindent
For $j = 1, \ldots, k$ and $i = 1, \ldots n_j$, $X_{i,j} \stackrel{ind}{\sim} N(\mu_j,\sigma^2)$. Let $\overline{X}_{\mbox{.}} = \sum_{j=1}^k\left(\frac{n_j}{n}\right) \overline{X}_j$. \\
$SSTO=SSB+SSW$, where $SSTO = \sum_{j=1}^k\sum_{i=1}^{n_j}(X_{i,j}-\overline{X}_{\mbox{.}})^2$, $SSB = \sum_{j=1}^k n_j(\overline{X}_j - \overline{X}_{\mbox{.}})^2$ and $SSW = \sum_{j=1}^k \sum_{i=1}^{n_j} (X_{i,j} - \overline{X}_j )^2$. $R^2 = \frac{SSB}{SSTO}$.  \vspace{1mm}

\noindent
Under $H_0: \mu_1 = \cdots \mu_k$, $F = \frac{SSB/(k-1)}{SSW/(n-k)} = \left(\frac{n-k}{k-1}\right) \left(\frac{R^2}{1-R^2}\right) \sim F(k-1,n-k)$.  \vspace{4mm}

\noindent
$\lambda(\mathbf{x}) = \frac{L(\widehat{\theta}_0,\mathbf{x})} {L(\widehat{\theta},\mathbf{x})}$, 
$C_k = \{\mathbf{x} \in S: \lambda(\mathbf{x}) \leq k \}$, where $0<k<1$. \\
$G^2_n = -2\ln\lambda(\mathbf{X}) \stackrel{d}{\rightarrow} Y \sim \chi^2(q)$, where $q$ is the number of $=$ signs in $H_0$.  \vspace{4mm}

\noindent
\begin{tabular}{lll}
$\pi(\theta|x) = \frac{f(x|\theta)\pi(\theta)}{\int f(x|t) \pi(t) \, dt}
 \propto f(x|\theta)\pi(\theta)$ & ~ &
$E[f(x|\Theta)|\mathbf{x}] = \int f(x|\theta) \, \pi(\theta|\mathbf{x}) \, d\theta$ \\
&&\\ % Space
$L(\theta,\mathbf{x}) = g[\theta, t(\mathbf{x})] \, h(\mathbf{x})$ & ~ &
$I(\theta) = E\left(\frac{\partial}{\partial\theta} \ln f(X|\theta) \right)^2
= -E\left(\frac{\partial^2}{\partial\theta^2} \ln f(X|\theta) \right)$            \\
&&\\ % Space
If $E(T) = \theta$, $Var(T) \geq \frac{1}{n\,I(\theta)}$ & ~ &
$\frac{\sqrt{n}\left(\widehat{\Theta}_n-\theta\right)}{\sqrt{1/I(\theta)}} 
\stackrel{d}{\rightarrow} Z \sim N(0,1)$ \hspace{3mm}
$\frac{\sqrt{n}\left(\widehat{\Theta}_n-\theta\right)}{\sqrt{1/I(\widehat{\Theta}_n)}} 
\stackrel{d}{\rightarrow} Z \sim N(0,1)$
\end{tabular}

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{center}
                    {\huge \hspace{20mm} R Code }
{\Large
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{lll} \\ \hline
\textbf{Distribution} & ~~~~~~~\textbf{CDF} $F(x|\theta)$ & ~~~~~\textbf{Quantile} \\ \hline
Exponential$(\lambda)$ & \texttt{pexp(2.7,2)} & \texttt{qexp(0.99,1)} \\
Gamma$(\alpha,\lambda)$ & \texttt{pgamma(1,shape=21,rate=23.35)} & 
\texttt{qgamma(0.025,4,23.25)} \\
Normal$(\mu,\sigma)$ & \texttt{pnorm(160,mean=100,sd=15)} & \texttt{qnorm(0.975,0,1)} \\
Chi-squared$(\nu)$ & \texttt{pchisq(1.7,df=4)} & \texttt{qchisq(0.95,1)} \\
$t(\nu)$ & \texttt{pt(2.14,df=10)} & \texttt{qt(0.975,df=134)} \\
$F(\nu_1,\nu_2)$ & \texttt{pf(3.17,df1=6,df2=114)} & \texttt{qf(0.95,6,114)} \\
Beta$(\alpha,\beta)$ & \texttt{pbeta(0.25,61,41)} &  \texttt{qbeta(0.5,61,41)} \\ \hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
} % End size
\end{center} \vspace{20mm}


%
\begin{comment}




\end{comment}

%%%%%%%%%%%%%%%%%%%%%% Tables at the end %%%%%%%%%%%%%%%%%%%%%%

\includepdf{NormalTable.pdf}
\includepdf{t-table.pdf}

\begin{center}
{\LARGE Quantiles of the chi-squared Distribution} \vspace{5mm}

\includegraphics[width=5.6in]{chisqtable}
\end{center}

\end{document}

&&\\ % Space
$F_{_X}(x) = P(X \leq x)$ & 
$p_{_X}(x) = \sum_y p_{_{X,Y}}(x,y)$ &
$f_{_X}(x) = \int_{-\infty}^\infty f_{_{X,Y}}(x,y) \, dy$ \\
&&\\ % Space
& $p_{_{Y|X}}(y|x) \stackrel{def}{=} \frac{p_{_{X,Y}}(x,y)}{p_{_X}(x)}$
& $f_{_{Y|X}}(y|x) \stackrel{def}{=} \frac{f_{_{X,Y}}(x,y)} {f_{_X}(x)}$ \\