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\begin{center}   
{\Large \textbf{STA 431s13 Formulas}}\footnote{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/312f12} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/431s13}}
}\\
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\begin{tabular}{ll}
$E(g(X)) = \int_{-\infty}^\infty g(x) \, f_{_X}(x) \, dx$, &
or $E(g(X)) =  \sum_x g(x) \, p_{_X}(x) $   \\ 
$Var(X) = E[(X-\mu_{_X})^2]$ &
$Cov(X,Y) = E[(X-\mu_{_X})(Y-\mu_{_Y})]$    \\
$Corr(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$ &
%$\boldsymbol{\Sigma} = \mathbf{P}\boldsymbol{\Lambda}\mathbf{P}^\prime$ 
\\
If $W=X+e$,  & Reliability is $Corr(W,X)^2 = \frac{\sigma^2_X}{\sigma^2_X+\sigma^2_e}$ \\
$V(\mathbf{X}) = 
E\left\{(\mathbf{X}-\boldsymbol{\mu}_x)(\mathbf{X}-\boldsymbol{\mu}_x)^\prime\right\}$ &
$C(\mathbf{X,Y}) = E\left\{ (\mathbf{X}-\boldsymbol{\mu}_x)
                             (\mathbf{Y}-\boldsymbol{\mu}_y)^\prime\right\}$ \\
\multicolumn{2}{l} {If $\mathbf{X} \sim N(\boldsymbol{\mu},\boldsymbol{\Sigma} )$, then 
$\mathbf{AX} \sim N(\mathbf{A}\boldsymbol{\mu},
                    \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\prime )$.} 
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\vspace{3mm}

\begin{displaymath}
\begin{array}{l}
f(\mathbf{x}_i|\boldsymbol{\mu,\Sigma}) = 
           \frac{1}{|\boldsymbol{\Sigma}|^{\frac{1}{2}} (2 \pi)^{\frac{k}{2}}} 
                \exp\left[ -\frac{1}{2} (\mathbf{x}_i-\boldsymbol{\mu})^\prime
                 \boldsymbol{\Sigma}^{-1}(\mathbf{x}_i-\boldsymbol{\mu})\right] \\ \\
L(\boldsymbol{\mu,\Sigma}) = 
    |\boldsymbol{\Sigma}|^{-n/2} (2\pi)^{-nk/2} 
    \exp -\frac{n}{2}\left\{ tr(\boldsymbol{\widehat{\Sigma}\Sigma}^{-1}) +
    (\overline{\mathbf{x}}-\boldsymbol{\mu})^\prime \boldsymbol{\Sigma}^{-1} 
    (\overline{\mathbf{x}}-\boldsymbol{\mu}) \right\}, \mbox{ where} \\ \\
\displaystyle{\widehat{\boldsymbol{\Sigma}} = \frac{1}{n} \sum_{i=1}^n 
\left(\mathbf{x}_i - \overline{\mathbf{x}} \right) 
\left(\mathbf{x}_i - \overline{\mathbf{x}} \right)^\prime}
\end{array}
\end{displaymath}

\begin{eqnarray*}
\mathbf{Y}_i &=& \boldsymbol{\beta} \mathbf{Y}_i + \boldsymbol{\Gamma} \mathbf{X}_i + \boldsymbol{\epsilon}_i \\
&& V(\mathbf{X}_i)=\boldsymbol{\Phi}_{11} \mbox{ and } 
   V(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi} \\ \\
\mathbf{F}_i &=& \left( \begin{array}{c}
         \mathbf{X}_i  \\ \mathbf{Y}_i 
             \end{array} \right)  \\
&& V(\mathbf{F}_i) = \boldsymbol{\Phi}
= \left( \begin{array}{c c}
V(\mathbf{X}_i)  & C(\mathbf{X}_i,\mathbf{Y}_i) \\
C(\mathbf{Y}_i,\mathbf{X}_i)  & V(\mathbf{Y}_i)
             \end{array} \right)
= \left( \begin{array}{c c}
         \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\
         \boldsymbol{\Phi}_{12}^\prime  & \boldsymbol{\Phi}_{22} \\
             \end{array} \right) \\ \\
\mathbf{D}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i \\
&& V(\mathbf{e}_i) = \boldsymbol{\Omega}
\end{eqnarray*}

\vspace{3mm}

\begin{center}
\textbf{Please see identifiability rules on the reverse.}
\end{center}

\includepdf[pages={-}]{LittleIdentRules}

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with $T$ the true score and $X$ observed,
