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\begin{center}   
{\Large \textbf{STA 442/2101 Formulas}}\\   % Version 2
\vspace{1 mm}
\end{center}

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


\noindent
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\begin{tabular}{lll}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MGF %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$M_Y(t) = E(e^{Yt})$ & ~~~~~ & $M_{aY}(t) = M_Y(at)$ \\
$M_{Y+a}(t) = e^{at}M_Y(t)$ & ~~~~~ & 
$M_{\sum_{i=1}^n Y_i}(t) = \prod_{i=1}^n M_{Y_i}(t)$
\\
$Y \sim N(\mu,\sigma^2)$ 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 $W_1\sim\chi^2(\nu_1)$} \\ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Linear Algebra %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{7 cm}{Columns of  $\mathbf{A}$ \emph{linearly dependent} means there is a vector $\mathbf{v} \neq \mathbf{0}$ with $\mathbf{Av} = \mathbf{0}$.} & ~~~~~ &
\parbox{7 cm}{Columns of  $\mathbf{A}$ \emph{linearly independent} means that $\mathbf{Av} = \mathbf{0}$ implies $\mathbf{v} = \mathbf{0}$.}
\\
\multicolumn{3}{l}{$\mathbf{A}$ \emph{positive definite} means  $\mathbf{v}^\top \mathbf{Av} > 0$ for all vectors $\mathbf{v} \neq \mathbf{0}$.} \\
$\boldsymbol{\Sigma} = \mathbf{P} \boldsymbol{\Lambda}\mathbf{P}^\top$
& ~~~~~ & 
$\boldsymbol{\Sigma}^{-1} = \mathbf{P} \boldsymbol{\Lambda}^{-1} \mathbf{P}^\top$
\\
$\boldsymbol{\Sigma}^{1/2} = \mathbf{P} \boldsymbol{\Lambda}^{1/2} \mathbf{P}^\top$
& ~~~~~ &
$\boldsymbol{\Sigma}^{-1/2} = \mathbf{P} \boldsymbol{\Lambda}^{-1/2} \mathbf{P}^\top$
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Random vectors and MVN %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$cov(\mathbf{w}) = 
E\left\{(\mathbf{w}-\boldsymbol{\mu}_w)(\mathbf{w}-\boldsymbol{\mu}_w)^\top\right\}$ 
& ~~~~~ & 
$C(\mathbf{w,t}) = E\left\{ (\mathbf{w}-\boldsymbol{\mu}_w)
                             (\mathbf{t}-\boldsymbol{\mu}_t)^\top\right\}$
\\
$cov(\mathbf{w}) = E\{\mathbf{ww}^\top\} - \boldsymbol{\mu}_w\boldsymbol{\mu}_w^\top$
& ~~~~~ &
$cov(\mathbf{Aw}) = \mathbf{A}cov(\mathbf{w}) \mathbf{A}^\top$
\\

%%%%%%%%%%%%%%%%%%%%%%%% MVN %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

If $\mathbf{w} \sim N_p(\boldsymbol{\mu},\boldsymbol{\Sigma} )$, then
$\mathbf{Aw}+\mathbf{c} \sim N_r(\mathbf{A}\boldsymbol{\mu} + \mathbf{c},
                    \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\top )$
& ~~~~~ & 
and $(\mathbf{w}-\boldsymbol{\mu})^\top
                 \boldsymbol{\Sigma}^{-1}(\mathbf{w}-\boldsymbol{\mu}) \sim \chi^2 (p)$
\\
%%%%%%%%%%%%%%%%%%%%%%%%%% Regression %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$Y_i = \beta_0 + \beta_1 x_{i,1} + \cdots + \beta_{p-1} x_{i,p-1} + \epsilon_i$
& ~~~~~ & 
$\epsilon_1, \ldots, \epsilon_n$ independent $N(0,\sigma^2)$
\\
$\mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$ 
& ~~~~~ &
$\boldsymbol{\epsilon} \sim N_n(\mathbf{0},\sigma^2\mathbf{I}_n)$
\\
$\widehat{\boldsymbol{\beta}} = (\mathbf{X}^\top \mathbf{X})^{-1} 
                   \mathbf{X}^\top \mathbf{y} \sim N_p\left(\boldsymbol{\beta}, 
                   \sigma^2 (\mathbf{X}^\top \mathbf{X})^{-1}\right)$
& ~~~~~ &
$\widehat{\mathbf{y}} = \mathbf{X}\widehat{\boldsymbol{\beta}} = \mathbf{Hy}$, where 
$\mathbf{H} = \mathbf{X}(\mathbf{X}^\top \mathbf{X})^{-1} 
                   \mathbf{X}^\top $
\\ 
$\textbf{e} = \mathbf{y} - \widehat{\mathbf{y}} = (\mathbf{I}-\mathbf{H})\mathbf{y}$ 
& ~~~~~ &
$\widehat{\boldsymbol{\beta}}$ and $\textbf{e}$ are independent under normality.
\\
$\frac{SSE}{\sigma^2} = \frac{\textbf{e}^\top \textbf{e}}{\sigma^2}  \sim \chi^2(n-p)$
& ~~~~~ &
$MSE = \frac{SSE}{n-p}$
\\
$\sum_{i=1}^n(Y_i-\overline{Y})^2 = \sum_{i=1}^n(Y_i-\widehat{Y}_i)^2 + \sum_{i=1}^n(\widehat{Y}_i-\overline{Y})^2$
& ~~~~~ &
$SST=SSE+SSR$ and $R^2 = \frac{SSR}{SST}$
\\
$T = \frac{Z}{\sqrt{W/\nu}} \sim t(\nu)$ 
& ~~~~~ & 
$F = \frac{W_1/\nu_1}{W_2/\nu_2} \sim F(\nu_1,\nu_2)$ 
\\
    $T = \frac{\mathbf{a}^\top \widehat{\boldsymbol{\beta}}-\mathbf{a}^\top \boldsymbol{\beta}}
             {\sqrt{MSE \, \mathbf{a}^\top 
             (\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{a}}} \sim t(n-p)$
& ~~~~~ &
$T = \frac{Y_{n+1}-\mathbf{x}_{n+1}^\top \widehat{\boldsymbol{\beta}}}
             {\sqrt{MSE \, (1+\mathbf{x}_{n+1}^\top 
             (\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{x}_{n+1})}} \sim t(n-p)$
\\
$F^* = \frac{SSR_F-SSR_R}{r \, MSE_F}  \sim F(r,n-p)$
& ~~~~~ &
$a = \frac{R^2_F-R^2_R}{1-R^2_R} = \frac{rF^*}{n-p+rF^*}$
\\
$F^* = \frac{(\mathbf{L}\widehat{\boldsymbol{\beta}}-\mathbf{h})^\top
            (\mathbf{L}(\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{L}^\top)^{-1}
            (\mathbf{L}\widehat{\boldsymbol{\beta}}-\mathbf{h})}
           {r \, MSE} \sim F(r,n-p)$
& ~~~~~ &
$F^* =  \left(\frac{n-p}{r}\right) \left(\frac{a}{1-a}\right) \sim F(r,n-p)$
\\
$r = \frac{\sum_{i=1}^n (x_i-\overline{x})(y_i-\overline{y})}
               {\sqrt{\sum_{i=1}^n (x_i-\overline{x})^2} \sqrt{\sum_{i=1}^n (y_i-\overline{y})^2}}$ 
& ~~~~~ &
\parbox{7 cm}{If $ \sqrt{n}\left( T_n- \theta \right) \stackrel{d}{\rightarrow} T \sim N(0,\sigma^2)$ then 
$\sqrt{n}\left( g(T_n)- g(\theta) \right) \stackrel{d}{\rightarrow}  
  Y \sim N\left(0,g^\prime(\theta)^2 \, \sigma^2\right)$.}
\\
$ Z_i \stackrel{ind}{\sim} N(\mu_i,1)  \Rightarrow  \sum_{i=1}^n Z_i^2 \sim \chi^2_{nc}(n,\lambda=\sum_{i=1}^n \mu_i^2)$
& ~~~~~ &
$\lambda = \frac{(\mathbf{L}\boldsymbol{\beta}-\mathbf{h})^\top
            (\mathbf{L}(\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{L}^\top)^{-1}
            (\mathbf{L}\boldsymbol{\beta}-\mathbf{h})}
           {\sigma^2}$
\\
\end{tabular}
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\vspace{20mm}


\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/302f16} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f16}}



\end{document}

If $ \sqrt{n}\left( T_n- \theta \right) \stackrel{d}{\rightarrow} T \sim N(0,\sigma^2)$ then 
$\sqrt{n}\left( g(T_n)- g(\theta) \right) \stackrel{d}{\rightarrow}  
    Y \sim N\left(0,g^\prime(\theta)^2 \, \sigma^2\right)$.

& ~~~~~ &
$\lambda = \frac{(\mathbf{L}\boldsymbol{\beta}-\mathbf{h})^\top
            (\mathbf{L}(\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{L}^\top)^{-1}
            (\mathbf{L}\boldsymbol{\beta}-\mathbf{h})}
           {\sigma^2}$
\\
$ Z_i \stackrel{ind}{\sim} N(\mu_i,1)  \Rightarrow  \sum_{i=1}^n Z_i^2 \sim \chi^2_{nc}(n,\lambda=\sum_{i=1}^n \mu_i^2)$
& ~~~~~ &
 $\lambda = \frac{\sum_{k=1}^p n_k (\mu_k-\mu_.)^2}{\sigma^2}$, where 
                 $\mu_. = \sum_{k=1}^p \frac{n_k}{n} \mu_k$
\\
$ \log\left(\frac{\pi_i}{1-\pi_i} \right) = 
    \beta_0 + \beta_1 x_{i,1} + \ldots + \beta_{p-1} x_{i,p-1}$
& ~~~~~ &
   $\pi_i  =  \frac{e^{\beta_0 + \beta_1 x_{i,1} + \ldots + \beta_{p-1} x_{i,p-1}}}
                    {1+e^{\beta_0 + \beta_1 x_{i,1} + \ldots + \beta_{p-1} x_{i,p-1}}}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$G^2 = -2 \log \left(   
           \frac{\max_{\theta \in \Theta_0} L(\theta)}
                {\max_{\theta \in \Theta} L(\theta)}
           \right)$
& ~~~~~ &
$W_n = (\mathbf{L}\widehat{\boldsymbol{\theta}}_n-\mathbf{h})^\top 
\left(\mathbf{L} \widehat{\mathbf{V}}_n \mathbf{L}^\top\right)^{-1} 
(\mathbf{L}\widehat{\boldsymbol{\theta}}_n-\mathbf{h})$ 
\\

\begin{verbatim}
> # Chi-squared critical values
> df = 1:6
> Critical_Value = qchisq(0.95,df)
> cbind(df,Critical_Value)
     df Critical_Value
[1,]  1       3.841459
[2,]  2       5.991465
[3,]  3       7.814728
[4,]  4       9.487729
[5,]  5      11.070498
[6,]  6      12.591587
\end{verbatim}