\documentclass[10pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts} % for \mathbb{R} The set of reals
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}
%\pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{center}   
{\Large \textbf{STA 302f15 Assignment Seven}}
%\footnote{Copyright information is at the end of the last page.}
\vspace{1 mm}
\end{center}

\noindent
These problems are preparation for the quiz, and are not to be handed in. As usual, \textbf{you might be asked to prove things that are not true}. In this case you should say why the statement is not always true. 

\begin{enumerate} 

\item Let the continuous random vectors $\mathbf{y_1}$ and $\mathbf{y_2}$ be independent. Show that their joint moment-generating function is the product of their moment-generating functions. Since $\mathbf{y_1}$ and $\mathbf{y_2}$ are continuous, you will integrate. It is okay to represent a multiple integral with a single integral sign.


\item Show that if $\mathbf{y} \sim N_p(\boldsymbol{\mu},\boldsymbol{\Sigma})$, with $\boldsymbol{\Sigma}$ positive definite, then $W = (\mathbf{y}-\boldsymbol{\mu})^\prime
           \boldsymbol{\Sigma}^{-1}(\mathbf{y}-\boldsymbol{\mu})$ 
has a chi-square distribution with $p$ degrees of freedom.

\item \label{chisq} Let $Y_1, \ldots, Y_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution. 
    \begin{enumerate}
        \item Show $Cov(\overline{Y},(Y_j-\overline{Y}))=0$ for $j=1, \ldots, n$.
        \item Show that $\overline{Y}$ and $S^2$ are independent.
        \item Show that 
\begin{displaymath}
    \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1),
\end{displaymath}
where $S^2 = \frac{\sum_{i=1}^n\left(Y_i-\overline{Y} \right)^2 }{n-1}$.
Hint: $\sum_{i=1}^n\left(Y_i-\mu \right)^2 = \sum_{i=1}^n\left(Y_i-\overline{Y} + \overline{Y} - \mu \right)^2 = \ldots$
    \end{enumerate}

\item Recall the definition of the $t$ distribution. If $Z\sim N(0,1)$, $W \sim \chi^2(\nu)$ and $Z$ and $W$ are independent, then $T = \frac{Z}{\sqrt{W/\nu}}$ is said to have a $t$ distribution with $\nu$ degrees of freedom, and we write $T  \sim t(\nu)$. As Question~\ref{chisq}, let $Y_1, \ldots, Y_n$ be random sample from a $N(\mu,\sigma^2)$ distribution. Show that $T = \frac{\sqrt{n}(\overline{Y}-\mu)}{S} \sim t(n-1)$. 

\item In the multiple linear regression model, let the columns of the $\mathbf{X}$ matrix be linearly independent. Either (a) show that $(\mathbf{X}^\prime\mathbf{X})^{-1/2}$ is symmetric, or (b) show by a simple numerical example that $(\mathbf{X}^\prime\mathbf{X})^{-1/2}$ may not be symmetric. 

 \item In the general linear regression model with normal error terms, what is the distribution of $\mathbf{y}$? 

\item You know that the least squares estimate of $\boldsymbol{\beta}$ is $\widehat{\boldsymbol{\beta}} = (\mathbf{X}^\prime \mathbf{X})^{-1} \mathbf{X}^\prime \mathbf{y}$. What is the distribution of $\widehat{\boldsymbol{\beta}}$  assuming normal error terms? Show the calculations.

\item Let $\widehat{\mathbf{y}}=\mathbf{X}\hat{\boldsymbol{\beta}}$. What is the distribution of $\widehat{\mathbf{y}}$ assuming normal error terms? Show the expected value and covariance matrix calculations.

\item Let the vector of residuals $\hat{\boldsymbol{\epsilon}} = \mathbf{y}-\widehat{\mathbf{y}}$. What is the distribution of $\hat{\boldsymbol{\epsilon}}$  assuming normal error terms? Show the calculations. Simplify both the expected value (which is zero) and the covariance matrix.

\item For the general linear regression model with normal error terms, show that the $n \times (k+1)$ matrix of covariances $C(\hat{\boldsymbol{\epsilon}},\widehat{\boldsymbol{\beta}}) = \mathbf{0} $. Why does this show that $SSE = \hat{\boldsymbol{\epsilon}}^\prime\hat{\boldsymbol{\epsilon}}$ and $\widehat{\boldsymbol{\beta}}$ are independent?

\item Calculate $C(\widehat{\boldsymbol{\epsilon}},\widehat{\mathbf{y}})$; show your work. Why should you have known this answer without doing the calculation, assuming normal error terms? Why does the assumption of normality matter?

\item For the general linear regression model with normal error terms, show that $\hat{\boldsymbol{\epsilon}}$ and $\overline{y}$ are independent.

\end{enumerate}

\vspace{10mm}

\noindent
\begin{center}\begin{tabular}{l}
\hspace{6in} \\ \hline
\end{tabular}\end{center}
This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, 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/302f15} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f15}}

\end{document}


