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\begin{center}   
{\Large \textbf{STA 302 Formulas}}\\   % Version 1
\vspace{1 mm}
\end{center}

% Spectral decomposition, linear independence.
% MGFs
% Random vectors
% Linear model
% Distribution facts, incl x2 addup?
% Test stats and CIs


\noindent
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{lll}
\multicolumn{3}{l}{Columns of  $\mathbf{A}$ \emph{linearly dependent} means there is a vector $\mathbf{v} \neq \mathbf{0}$ with $\mathbf{Av} = \mathbf{0}$.} \\
\multicolumn{3}{l}{Columns of  $\mathbf{A}$ \emph{linearly independent} means that $\mathbf{Av} = \mathbf{0}$ implies $\mathbf{v} = \mathbf{0}$.} \\ 
$\boldsymbol{\Sigma} = \mathbf{CD} \mathbf{C}^\prime$
& ~~~~~ & 
$\boldsymbol{\Sigma}^{-1} = \mathbf{C} \mathbf{D}^{-1} \mathbf{C}^\prime$
\\
$\boldsymbol{\Sigma}^{1/2} = \mathbf{CD}^{1/2} \mathbf{C}^\prime$
& ~~~~~ &
$\boldsymbol{\Sigma}^{-1/2} = \mathbf{CD}^{-1/2} \mathbf{C}^\prime$
\\
$M_Y(t) = E(e^{Yt})$ & ~~~~~ & $M_{aY}(t) = M_Y(at)$ \\
$M_{Y+a}(t) = e^{at}M_Y(t)$ & ~~~~~ & 
$M_{\sum_{i=1}^n Y_i}(t) = \prod_{i=1}^n M_{Y_i}(t)$
\\
$Y \sim N(\mu,\sigma^2)$ means $M_Y(t) = e^{\mu t + \frac{1}{2}\sigma^2t^2}$
& ~~~~~ & 
$Y \sim \chi^2(\nu)$ means $M_Y(t) = (1-2t)^{-\nu/2}$
\\
\multicolumn{3}{l}{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 $W_1\sim\chi^2(\nu_1)$} \\ 
$cov(\mathbf{Y}) = 
E\left\{(\mathbf{Y}-\boldsymbol{\mu}_y)(\mathbf{Y}-\boldsymbol{\mu}_y)^\prime\right\}$ 
& ~~~~~ & 
$C(\mathbf{Y,T}) = E\left\{ (\mathbf{Y}-\boldsymbol{\mu}_y)
                             (\mathbf{T}-\boldsymbol{\mu}_t)^\prime\right\}$
\\
$cov(\mathbf{Y}) = E\{\mathbf{YY}^\prime\} - \boldsymbol{\mu}_y\boldsymbol{\mu}_y^\prime$
& ~~~~~ &
$cov(\mathbf{AY}) = \mathbf{A}cov(\mathbf{Y}) \mathbf{A}^\prime$
\\
$M_{\mathbf{Y}}(\mathbf{t}) = E(e^{\mathbf{t}^\prime\mathbf{Y}})$ 
& ~~~~~ & 
$M_{\mathbf{AY}}(\mathbf{t}) = M_{\mathbf{Y}}(\mathbf{A}^\prime\mathbf{t})$
\\
$M_{\mathbf{Y}+\mathbf{c}}(\mathbf{t}) = e^{\mathbf{t}^\prime\mathbf{c}} M_{\mathbf{Y}}(\mathbf{t})$
 & ~~~~~ & 
$\mathbf{Y} \sim N_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ means $M_{\mathbf{Y}}(\mathbf{t}) = e^{\mathbf{t}^\prime\boldsymbol{\mu} + \frac{1}{2} \mathbf{t}^\prime \boldsymbol{\Sigma} \mathbf{t}}$
\\
\multicolumn{3}{l}{$\mathbf{Y}_1$ and $\mathbf{Y}_2$ are independent if and only if
$M_{(\mathbf{Y}_1,\mathbf{Y}_2)^\prime}\left((\mathbf{t}_1,\mathbf{t}_2)^\prime\right)
= M_{\mathbf{Y}_1}(\mathbf{t}_1) M_{\mathbf{Y}_2}(\mathbf{t}_2)$} \\
If $\mathbf{Y} \sim N_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, then $\mathbf{AY} \sim N_q(\mathbf{A}\boldsymbol{\mu}, \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\prime)$,
 & ~~~~~ &
and $W = (\mathbf{Y}-\boldsymbol{\mu})^\prime
           \boldsymbol{\Sigma}^{-1}(\mathbf{Y}-\boldsymbol{\mu}) \sim \chi^2(p)$
\\
$r_{xy} = \frac{\sum_{i=1}^n (X_i-\overline{X})(Y_i-\overline{Y})}
               {\sqrt{\sum_{i=1}^n (X_i-\overline{X})^2} \sqrt{\sum_{i=1}^n (Y_i-\overline{Y})^2}}$
& ~~~~~ &
\\
$Y_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_k x_{ik} + \epsilon_i$
& ~~~~~ & 
$\epsilon_1, \ldots, \epsilon_n$ independent $N(0,\sigma^2)$
\\
$\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$ 
& ~~~~~ &
$\boldsymbol{\epsilon} \sim N_n(\mathbf{0},\sigma^2\mathbf{I}_n)$
\\
$\widehat{\boldsymbol{\beta}} = (\mathbf{X}^\prime \mathbf{X})^{-1} 
                   \mathbf{X}^\prime \mathbf{Y} $
& ~~~~~ &
$\widehat{\mathbf{Y}} = \mathbf{X}\widehat{\boldsymbol{\beta}} = \mathbf{HY}$, where 
$\mathbf{H} = \mathbf{X}(\mathbf{X}^\prime \mathbf{X})^{-1} 
                   \mathbf{X}^\prime $
\\ 
$\sum_{i=1}^n(Y_i-\overline{Y})^2 = \sum_{i=1}^n(Y_i-\widehat{Y}_i)^2 + \sum_{i=1}^n(\widehat{Y}_i-\overline{Y})^2$
& ~~~~~ &
$SST=SSE+SSR$ and $R^2 = \frac{SSR}{SST}$
\\
$\widehat{\boldsymbol{\epsilon}} = \mathbf{Y} - \widehat{\mathbf{Y}}$ 
& ~~~~~ &
$\widehat{\boldsymbol{\beta}} \sim N_{k+1}\left(\boldsymbol{\beta}, 
                   \sigma^2 (\mathbf{X}^\prime \mathbf{X})^{-1}\right)$
\\
$\widehat{\boldsymbol{\beta}}$ and $\widehat{\boldsymbol{\epsilon}}$ are independent under normality.
& ~~~~~ &
$SSE/\sigma^2 = \hat{\boldsymbol{\epsilon}}^\prime \hat{\boldsymbol{\epsilon}}/\sigma^2  \sim \chi^2(n-k-1)$ 
\\
$T = \frac{Z}{\sqrt{W/\nu}} \sim t(\nu)$
& ~~~~~ &
$F = \frac{W_1/\nu_1}{W_2/\nu_2} \sim F(\nu_1,\nu_2)$
\\
    $T = \frac{\mathbf{a}^\prime \widehat{\boldsymbol{\beta}}-\mathbf{a}^\prime \boldsymbol{\beta}}
             {s \, \sqrt{\mathbf{a}^\prime 
             (\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{a}}} \sim t(n-k-1)$
& ~~~~~ &
$T = \frac{Y_0-\mathbf{x}_0^\prime \widehat{\boldsymbol{\beta}}}
             {s \, \sqrt{(1+\mathbf{x}_0^\prime 
             (\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{x}_0)}} \sim t(n-k-1)$
\\
\multicolumn{3}{l}{$F = \frac{(\mathbf{C}\widehat{\boldsymbol{\beta}}-\mathbf{t})^\prime
            (\mathbf{C}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{C}^\prime)^{-1}
            (\mathbf{C}\widehat{\boldsymbol{\beta}}-\mathbf{t})}
           {q \, s^2} 
           = \frac{SSR-SSR(reduced)}{q \, s^2} \sim F(q,n-k-1)$, 
           where $s^2 = MSE = \frac{SSE}{n-k-1}$} \\
\end{tabular}
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%\vspace{10mm}


\noindent
\begin{center}\begin{tabular}{l}
\hspace{6.5in} \\ \hline
\end{tabular}\end{center}
This formula sheet was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistics, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/302f13} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f13}}



\end{document}


\noindent
~~If $\mathbf{Y} \sim N_p(\boldsymbol{\mu},\boldsymbol{\Sigma} )$, then 
$\mathbf{AY} \sim N_r(\mathbf{A}\boldsymbol{\mu},
                    \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\prime )$.

\vspace{3mm}


\noindent
~~If $E(\mathbf{Y})=\boldsymbol{\mu}$, then $V(\mathbf{Y})$ is defined by 
$V(\mathbf{Y}) = 
E\left\{(\mathbf{Y}-\boldsymbol{\mu})(\mathbf{Y}-\boldsymbol{\mu})^\prime\right\}$.

\vspace{3mm}


\multicolumn{3}{l}{If $X \sim N(\mu,\sigma^2)$, then $\frac{X^2}{\sigma^2} \sim \chi^2(1,\lambda)$, with
$\lambda = \frac{\mu^2}{\sigma^2}$} \\

$\phi = \frac{(\mathbf{L}\boldsymbol{\beta}-\mathbf{h})^\prime
            (\mathbf{L}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{L}^\prime)^{-1}
            (\mathbf{L}\boldsymbol{\beta}-\mathbf{h})}
           {\sigma^2}$
& ~~~~~ &
 \\
$f(y|\theta,\phi) = 
    \exp\left\{ \frac{y\theta-b(\theta)}{\phi} + c(y,\phi)\right\}$
& ~~~~~ &
$\theta = g(\mu) = \eta = \mathbf{x}^\prime\boldsymbol{\beta}$ \\