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\begin{center}   
{\Large \textbf{STA 305 Formulas}}\\   % Version 3
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\noindent
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$Var(Y) = E\{(Y-\mu_y)^2\}$
& ~~~~~ & 
$Cov(Y,T) = E\{(Y-\mu_y)(T-\mu_t)\}$
\\
$cov(\mathbf{Y}) = 
E\left\{(\mathbf{Y}-\boldsymbol{\mu}_y)(\mathbf{Y}-\boldsymbol{\mu}_y)^\prime\right\}$ 
& ~~~~~ & 
$C(\mathbf{Y,T}) = E\left\{ (\mathbf{Y}-\boldsymbol{\mu}_y)
                             (\mathbf{T}-\boldsymbol{\mu}_t)^\prime\right\}$
\\
\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 $W_1\sim\chi^2(\nu_1)$} \\
$T = \frac{Z}{\sqrt{W/\nu}} \sim t(\nu)$
& ~~~~~ &
$F = \frac{W_1/\nu_1}{W_2/\nu_2} \sim F(\nu_1,\nu_2)$
\\
\multicolumn{3}{l}{For the multivariate normal distribution \emph{only}, zero covariance implies independence.} \\
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 $W = (\mathbf{Y}-\boldsymbol{\mu})^\prime
           \boldsymbol{\Sigma}^{-1}(\mathbf{Y}-\boldsymbol{\mu}) \sim \chi^2(p)$
\\
$Y_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_{p-1} x_{i,p-1} + \epsilon_i$
& ~~~~~ & 
$\epsilon_1, \ldots, \epsilon_n$ 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 $R^2 = \frac{SSR}{SST}$
\\
$\widehat{\boldsymbol{\epsilon}} = \mathbf{Y} - \widehat{\mathbf{Y}}$ 
& ~~~~~ &
$\widehat{\boldsymbol{\beta}} \sim N_p\left(\boldsymbol{\beta}, 
                   \sigma^2 (\mathbf{X}^\prime \mathbf{X})^{-1}\right)$
\\
$\widehat{\boldsymbol{\beta}}$ and $\widehat{\boldsymbol{\epsilon}}$ are independent under normality.
& ~~~~~ &
$SSE/\sigma^2 = \hat{\boldsymbol{\epsilon}}^\prime \hat{\boldsymbol{\epsilon}}/\sigma^2  \sim \chi^2(n-p)$ 
\\
    $T = \frac{\mathbf{a}^\prime \widehat{\boldsymbol{\beta}}-\mathbf{a}^\prime \boldsymbol{\beta}}
             {\sqrt{MSE \, \mathbf{a}^\prime 
             (\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{a}}} \sim t(n-p)$
& ~~~~~ &
$F^* = \frac{(\mathbf{C}\widehat{\boldsymbol{\beta}}-\mathbf{t})^\prime
            (\mathbf{C}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{C}^\prime)^{-1}
            (\mathbf{C}\widehat{\boldsymbol{\beta}}-\mathbf{t})}
           {q \, MSE}  \sim F(q,n-p,\lambda)$
\\
$F^* =  \frac{SSR-SSR(reduced)}{q \, MSE} \sim F(q,n-p,\lambda)$
& ~~~~~ & where $MSE = \frac{SSE}{n-p}$  \\
$ \lambda = \frac{(\mathbf{C}\boldsymbol{\beta}-\mathbf{t})^\prime
            (\mathbf{C}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{C}^\prime)^{-1}
            (\mathbf{C}\boldsymbol{\beta}-\mathbf{t})}
           {\sigma^2}$
& ~~~~~ &
\\
\multicolumn{3}{l}{Simple random sample of $n$ units from $N$ without replacement. $Z_i=1$ if unit $i$ is chosen, zero otherwise.} \\
$E(Z_i)= P(Z_i=1)= \frac{n}{N}$ 
& ~~~~~ &
$\overline{y}_u = \frac{1}{N}\sum_{i=1}^N y_i$
 \\
$\overline{y} = \frac{1}{n}\sum_{i=1}^N Z_i y_i$
& ~~~~~ &
$S^2 = \frac{1}{N-1}\sum_{i=1}^N (y_i-\overline{y}_u)^2$
\\
$c = a_1\mu_1 + a_2\mu_2 + \cdots + a_p\mu_p$
& ~~~~~ &
$\widehat{c} = a_1\overline{Y}_1 + a_2\overline{Y}_2 + \cdots + a_p\overline{Y}_p$
\\
$Pr\left\{ \cup_{j=1}^k A_j \right\} \leq \sum_{j=1}^k Pr\{A_j\}$
& ~~~~~ &
Reject $H_0$ with a Scheff\'e test if $F_2 > \frac{q}{s}f_\alpha(q,n-p)$
\\
$n =  \frac{\sigma^2 \, z_{\alpha/2}^2 \sum_{j=1}^p\frac{a_j^2}{f_j} }{m^2}$
& ~~~~~ &
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$1-\alpha$     & 0.80 & 0.90 & 0.95 & 0.99 \\
$z_{\alpha/2}$ & 1.28 & 1.64 & 1.96 & 2.58
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\noindent
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\hspace{6.5in} \\ \hline
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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/305s14} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/305s14}}



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\begin{tabular}{ccccc}
$1-\alpha$     & 0.80 & 0.90 & 0.95 & 0.99 \\
$z_{\alpha/2}$ & 1.28 & 1.64 & 1.96 & 2.58
\end{tabular}


\multicolumn{3}{l}{Columns of  $\mathbf{A}$ \emph{linearly independent} means that $\mathbf{Av} = \mathbf{0}$ implies $\mathbf{v} = \mathbf{0}$.} \\ 
$\boldsymbol{\Sigma} = \mathbf{CD} \mathbf{C}^\prime$