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{\Large \textbf{STA 431s17 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/431s17} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/431s17}}
}\\
\vspace{3 mm}
\noindent
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\begin{tabular}{ll}
$Var(X) = E\left\{(X-\mu_{_x})^2 \right\} = E(X^2)-\mu_x^2$ &
$Cov(X,Y) = E\left\{ (X-\mu_{_x})(Y-\mu_{_y}) \right\} = E(XY)-\mu_x\mu_y$ \\
$Corr(X,Y) = \rho_{xy} = \frac{Cov(X,Y)}{\sigma_x\sigma_y}$ & 
$r_{xy} = \frac{\sum_{i=1}^n (x_i-\overline{x})(y_i-\overline{y})}
               {\sqrt{\sum_{i=1}^n (x_i-\overline{x})^2} \sqrt{\sum_{i=1}^n (y_i-\overline{y})^2}}$ \\
$cov(\mathbf{X}) = 
E\left\{(\mathbf{X}-\boldsymbol{\mu}_x)(\mathbf{X}-\boldsymbol{\mu}_x)^\top\right\}$ &
$cov(\mathbf{X,Y}) = E\left\{ (\mathbf{X}-\boldsymbol{\mu}_x)
                             (\mathbf{Y}-\boldsymbol{\mu}_y)^\top\right\}$ \\
$cov(\mathbf{AX}) = \mathbf{A}\boldsymbol{\Sigma}_x\mathbf{A}^\top$ &
$cov(\mathbf{AX},\mathbf{BX}) = \mathbf{A}\boldsymbol{\Sigma}_x\mathbf{B}^\top$
\\
$\mathbf{L} = \mathbf{A}_1\mathbf{X}_1 + \cdots +  \mathbf{A}_m\mathbf{X}_m  + \mathbf{b}$ &
$\stackrel{c}{\mathbf{L}} = \mathbf{A}_1 \stackrel{c}{\mathbf{X}}_1 + \cdots + \mathbf{A}_m \stackrel{c}{\mathbf{X}}_m$  \\
$cov(\mathbf{L}) =  E(\stackrel{c}{\mathbf{L}}\stackrel{c}{\mathbf{L}}
\stackrel{\top}{\vphantom{r}})$ &
$cov(\mathbf{L}_1,\mathbf{L}_2) = E(\stackrel{c}{\mathbf{L}}_1\,\stackrel{c}{\mathbf{L}}
    \stackrel{\top}{\vphantom{r}_2})$ \\
$f(x|\mu,\sigma^2) =  \frac{1}{\sigma \sqrt{2\pi}} 
                  \exp \left\{-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right\}$ &
$f(\mathbf{x}|\boldsymbol{\mu,\Sigma}) = 
            \frac{1}{|\boldsymbol{\Sigma}|^{\frac{1}{2}} (2 \pi)^{\frac{p}{2}}} 
                \exp\left\{ -\frac{1}{2} (\mathbf{x}-\boldsymbol{\mu})^\top
                 \boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right\}$ \\
\multicolumn{2}{l} {If $\mathbf{X} \sim N(\boldsymbol{\mu},\boldsymbol{\Sigma} )$, then 
$\mathbf{AX} + \mathbf{b} \sim N_p(\mathbf{A}\boldsymbol{\mu} + \mathbf{b},
                    \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\top )$.} \\
\multicolumn{2}{l} {$L(\boldsymbol{\mu,\Sigma}) =     |\boldsymbol{\Sigma}|^{-n/2} (2\pi)^{-np/2} 
    \exp -\frac{n}{2}\left\{ tr(\boldsymbol{\widehat{\Sigma}\Sigma}^{-1}) +
    (\overline{\mathbf{x}}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} 
    (\overline{\mathbf{x}}-\boldsymbol{\mu}) \right\}$   } \\
    $\boldsymbol{\widehat{\Sigma}} = 
    \frac{1}{n}\sum_{i=1}^n (\mathbf{x}_i-\overline{\mathbf{x}}) 
                        (\mathbf{x}_i-\overline{\mathbf{x}})^\top $ & 
$G^2 = -2 \ln \left( \frac{\max_{\theta \in \Theta_0} L(\theta)}
                          {\max_{\theta \in \Theta}   L(\theta)  }  \right) 
     = -2 \ln \left( \frac{L(\widehat{\theta}_0)}{  L(\widehat{\theta})  }\right)$  \\
If $W=X+e$,  & Reliability is $Corr(W,X)^2 = \frac{\sigma^2_x}{\sigma^2_x+\sigma^2_e}$ \\
% & \\ % To make a space
\multicolumn{2}{c} {\sffamily The Double Measurement Model in centered form:~~~~~~~~~~~~} \\
$\mathbf{Y}_i = 
\boldsymbol{\beta} \mathbf{X}_i +  \boldsymbol{\epsilon}_i$ &
$cov(\mathbf{X}_i)=\boldsymbol{\Phi}_x$, $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi}$ \\
${\mathbf{F}}_i = \left( \begin{array}{c}
{\mathbf{X}}_i  \\ {\mathbf{Y}}_i \end{array} \right)$ &
\hspace{-2.4mm}\begin{tabular}{l}
$\mathbf{X}_i$ is $p \times 1$, $\mathbf{Y}_i$ is $q \times 1$, $\mathbf{F}_i$ 
is $(p+q) \times 1$ \\
$cov(\mathbf{F}_i) = \boldsymbol{\Phi}$
\end{tabular} \\
${\mathbf{D}}_{i,1} = {\mathbf{F}}_i + \mathbf{e}_{i,1}$ &
$cov(\mathbf{e}_{i,1})=\boldsymbol{\Omega}_1$, 
$cov(\mathbf{e}_{i,2})=\boldsymbol{\Omega}_2$ \\
${\mathbf{D}}_{i,2} = {\mathbf{F}}_i + \mathbf{e}_{i,2}$ &
$\mathbf{X}_i$, $\boldsymbol{\epsilon}_i$, $\mathbf{e}_{i,1}$ and 
$\mathbf{e}_{i,2}$ are independent. \\
\multicolumn{2}{c} {\sffamily The General Structural Equation Model in centered form: ~~~~~~} \\
$\mathbf{Y}_i = \boldsymbol{\beta} \mathbf{Y}_i + \boldsymbol{\Gamma} \mathbf{X}_i + \boldsymbol{\epsilon}_i$ &
$cov(\mathbf{X}_i)=\boldsymbol{\Phi}_x$ and $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi}$\\
$\mathbf{F}_i = \left( \begin{array}{c}
         \mathbf{X}_i  \\ \mathbf{Y}_i 
             \end{array} \right)$ &
$cov(\mathbf{F}_i) = \boldsymbol{\Phi}
= \left( \begin{array}{c c}
         \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\
         \boldsymbol{\Phi}_{12}^\top  & \boldsymbol{\Phi}_{22} \\
             \end{array} \right)$ \\
$\mathbf{D}_i = \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i$ &
$cov(\mathbf{e}_i) = \boldsymbol{\Omega}$ \\
$\mathbf{X}_i$, $\boldsymbol{\epsilon}_i$ and $\mathbf{e}_i$ are independent. &
$\mathbf{X}_i$ is $p \times 1$, $\mathbf{Y}_i$ is $q \times 1$, $\mathbf{D}_i$ is $k \times 1$. \\
$\boldsymbol{\Phi}_x$ and $\boldsymbol{\Psi}$ are positive definite. & \\
% Not Omega because of the trick of adding observed variables.
\end{tabular}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{verbatim}
> df = 1:10; CriticalValue = qchisq(0.95,df); cbind(df,CriticalValue)
      df CriticalValue
 [1,]  1      3.841459
 [2,]  2      5.991465
 [3,]  3      7.814728
 [4,]  4      9.487729
 [5,]  5     11.070498
 [6,]  6     12.591587
 [7,]  7     14.067140
 [8,]  8     15.507313
 [9,]  9     16.918978
[10,] 10     18.307038
\end{verbatim}