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

\noindent
\begin{tabular}{cccl}
$Z_1 = \frac{\sqrt{n}(\overline{Y}_n-\pi_0)}{\sqrt{\pi_0(1-\pi_0)}}$ & 
$Z_2 = \frac{\sqrt{n}(\overline{Y}_n-\pi_0)}{\sqrt{\overline{Y}_n(1-\overline{Y}_n)}}$ &
$\overline{Y}_n \pm z_{\alpha/2}\sqrt{\frac{\overline{Y}_n(1-\overline{Y}_n)}{n}}$ &
\begin{minipage}{3in}
\begin{verbatim}
> qnorm(0.975)
[1] 1.959964
> qnorm(0.995)
[1] 2.575829
\end{verbatim}
\end{minipage}
\end{tabular}

\vspace{3mm}

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

\vspace{3mm}

\noindent
~~If $\mathbf{X} \sim N(\boldsymbol{\mu},\boldsymbol{\Sigma} )$, then 
$\mathbf{AX} \sim N(\mathbf{A}\boldsymbol{\mu},
                    \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\prime )$.

\vspace{3mm}


\noindent
~~If $\lim_{n \rightarrow \infty} E(T_n) = \theta$  and $\lim_{n \rightarrow \infty} Var(T_n) = 0$, then $T_n \stackrel{P}{\rightarrow} \theta$

\vspace{3mm}


\noindent
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\begin{tabular}{lll} 
$\mathbf{Y}_n = \sqrt{n}(\overline{\mathbf{X}}_n-\boldsymbol{\mu}) 
\stackrel{d}{\rightarrow} \mathbf{Y} \sim N(\mathbf{0},\boldsymbol{\Sigma})$
& ~~~~~ & 
$\sqrt{n}(g(\mathbf{T}_n)-g(\boldsymbol{\theta})) \stackrel{d}{\rightarrow} 
\mbox{\.{g}} (\boldsymbol{\theta}) \mathbf{T}$, ~~\.{g}$(\mathbf{x}) = 
\left[ \frac{\partial g_i}{\partial x_j} \right]_{k \times d}$
\\
$P(n_1, \ldots, n_c) = \binom{n}{n_1~\cdots~n_c} \pi_1^{n_1} \cdots \pi_c^{n_c}$ 
& ~~~~~ &
$L(\boldsymbol{\pi}) =  \prod_{i=1}^n \pi_1^{y_{i,1}} \pi_2^{y_{i,2}} \cdots \pi_c^{y_{i,c}}
                        = \pi_1^{n_1} \pi_2^{n_2} \cdots \pi_c^{n_c}$ \\ 
$G^2 = -2 \log \left(   
           \frac{\max_{\theta \in \Theta_0} L(\theta)}
                {\max_{\theta \in \Theta} L(\theta)}
           \right)$
& ~~~~~ &
$W_n = (\mathbf{L}\widehat{\boldsymbol{\theta}}_n-\mathbf{h})^\prime 
\left(\mathbf{L} \widehat{\mathbf{V}}_n \mathbf{L}^\prime\right)^{-1} 
(\mathbf{L}\widehat{\boldsymbol{\theta}}_n-\mathbf{h})$ \\
\multicolumn{3}{l}{If $X \sim N(\mu,\sigma^2)$, then $\frac{X^2}{\sigma^2} \sim \chi^2(1,\lambda)$, with
$\lambda = \frac{\mu^2}{\sigma^2}$} \\
$\widehat{\boldsymbol{\beta}} = (\mathbf{X}^\prime \mathbf{X})^{-1} 
                   \mathbf{X}^\prime \mathbf{Y} \sim N_p\left(\boldsymbol{\beta}, 
                   \sigma^2 (\mathbf{X}^\prime \mathbf{X})^{-1}\right)$
& ~~~~~ &
$SSE/\sigma^2 \sim \chi^2(n-p)$ \\
\multicolumn{3}{l}{$F = \frac{(\mathbf{L}\widehat{\boldsymbol{\beta}}-\mathbf{h})^\prime
            (\mathbf{L}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{L}^\prime)^{-1}
            (\mathbf{L}\widehat{\boldsymbol{\beta}}-\mathbf{h})}
           {r \, MSE_F} 
           = \frac{(SSR_F-SSR_R)/r}{MSE_F} 
           = \left(\frac{n-p}{r}\right) \left(\frac{a}{1-a}\right)$,
           where $a = \frac{R^2_F-R^2_R}{1-R^2_R} = \frac{rF}{n-p+rF}$} \\
$\phi = \frac{(\mathbf{L}\boldsymbol{\beta}-\mathbf{h})^\prime
            (\mathbf{L}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{L}^\prime)^{-1}
            (\mathbf{L}\boldsymbol{\beta}-\mathbf{h})}
           {\sigma^2}$
& ~~~~~ &
 \\
$f(y|\theta,\phi) = 
    \exp\left\{ \frac{y\theta-b(\theta)}{\phi} + c(y,\phi)\right\}$
& ~~~~~ &
$\theta = g(\mu) = \eta = \mathbf{x}^\prime\boldsymbol{\beta}$ \\


\end{tabular}
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\vspace{10mm}

\begin{verbatim}
> df = 1:8
> CriticalValue = qchisq(0.95,df)
> round(rbind(df,CriticalValue),3)
               [,1]  [,2]  [,3]  [,4]  [,5]   [,6]   [,7]   [,8]
df            1.000 2.000 3.000 4.000  5.00  6.000  7.000  8.000
CriticalValue 3.841 5.991 7.815 9.488 11.07 12.592 14.067 15.507
\end{verbatim}





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