% 302f13Assignment2.tex      MGFs and one regression through the origin
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{\Large \textbf{STA 302f13 Assignment Two}}\footnote{Copyright information is at the end of the last page.}
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\noindent
These problems are preparation for the quiz in tutorial on Friday September 27th, and are not to be handed in. Starting with Problem~\ref{mgfstart}, you can play a little game. Try not to do the same work twice. Instead, use results of earlier problems whenever possible. 

\begin{enumerate} 


    \item Sometimes, you want the least squares line to go through the origin, so that predicted $Y$ automatically equals zero when $x=0$. For example, suppose the cases are half-kilogram batches of rice purchased from grocery stores. The independent variable $x$ is concentration of arsenic in the rice before washing, and the dependent variable $Y$ is concentration of arsenic after washing. Discounting the very unlikely possibility that arsenic contamination can happen \emph{during} washing, you want to use your knowledge that zero arsenic before washing implies zero arsenic after washing. You will use your knowledge by building it into the statistical model. 

Accordingly, let $Y_i = \beta x_i + \epsilon_i$
for $i=1, \ldots, n$, where  $\epsilon_1, \ldots, \epsilon_n$ are a random sample from a distribution with expected value zero and variance $\sigma^2$, and $\beta$ and $\sigma^2$ are unknown constants. The numbers $x_1, \ldots, x_n$ are known, observed constants. 
        \begin{enumerate}
            \item What is $E(Y_i)$?
            \item What is $Var(Y_i)$?
            \item Find the Least Squares estimate of $\beta$ by minimizing
                  the function 
                  \begin{displaymath}  
                      Q(\beta)=\sum_{i=1}^n(Y_i-\beta x_i)^2
                  \end{displaymath}                  
                  over all values of $\beta$. Let $\widehat{\beta}$ denote the point at which $Q(\beta)$ is minimal.
            \item Give the equation of the least-squares line. Of course it's the \emph{constrained} least-squares line, passing through $(0,0)$.
            \item Recall that a statistic is an \emph{unbiased estimator} of a parameter if the expected value of the statistic is equal to the parameter.  Is $\widehat{\beta}$ an unbiased estimator of $\beta$? Answer Yes or No and show your work.
            \item Let 
            $\widehat{\beta}_2 = \frac{\overline{Y}_n}{\overline{x}_n}$. Is $\widehat{\beta}_2$ also unbiased for $\beta$? Answer Yes or No  and show your work.

% Need numerical beta-hat and predicted Y here. Maybe a set of (x,Y) values. 

            \item \emph{This last part is a challenge for your entertainment. It will not be on the quiz or the final exam.} Prove that $\widehat{\beta}$ is a more accurate estimator than $\widehat{\beta}_{2}$ in the sense that it has smaller variance. Hint: The sample variance of the independent variable values cannot be negative. 
        \end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%% MGF %%%%%%%%%%%%%%%%%%%%%%%%%

    \item \label{mgfstart} Denote the moment-generating function of a random variable $Y$ by $M_Y(t)$. The moment-generating function is defined by $M_Y(t) = E(e^{Yt})$.
    \begin{enumerate}
        \item Let $a$ be a constant. Prove that $M_{aX}(t) = M_X(at)$.
        \item Prove that $M_{X+a}(t) = e^{at}M_X(t)$.
        \item Let $X_1, \ldots, X_n$ be \emph{independent} random variables. Prove that 
\begin{displaymath}
    M_{\sum_{i=1}^n X_i}(t) = \prod_{i=1}^n M_{X_i}(t).
\end{displaymath}
For convenience, you may assume that $X_1, \ldots, X_n$ are all continuous, so you will integrate.
    \end{enumerate} 

\newpage    

     \item Recall that if $X\sim N(\mu,\sigma^2)$, it has moment-generating function $M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2t^2}$. 
    \begin{enumerate}
        \item Let $X\sim N(\mu,\sigma^2)$ and $Y=aX+b$, where $a$ and $b$ are constants. Find the distribution of $Y$. Show your work. 
        \item Let $X\sim N(\mu,\sigma^2)$ and $Z = \frac{X-\mu}{\sigma}$. Find the distribution of $Z$.         
        \item Let $X_1, \ldots, X_n$ be random sample from a $N(\mu,\sigma^2)$ distribution. Find the distribution of $Y = \sum_{i=1}^nX_i$. 
        \item Let $X_1, \ldots, X_n$ be random sample from a $N(\mu,\sigma^2)$ distribution. Find the distribution of the sample mean $\overline{X}$.
        \item Let $X_1, \ldots, X_n$ be random sample from a $N(\mu,\sigma^2)$ distribution. Find the distribution of $Z = \frac{\sqrt{n}(\overline{X}-\mu)}{\sigma}$. 
    \end{enumerate}

    \item A Chi-squared random variable $X$ with parameter $\nu>0$ has moment-generating function $M_X(t) = (1-2t)^{-\nu/2}$. 
    \begin{enumerate}
        \item Let $X_1, \ldots, X_n$ be independent random variables with $X_i \sim \chi^2(\nu_i)$ for $i=1, \ldots, n$. Find the distribution of $Y = \sum_{i=1}^n X_i$. 
        \item Let $Z \sim N(0,1)$. Find the distribution of $Y=Z^2$. 
        \item Let $X_1, \ldots, X_n$ be random sample from a $N(\mu,\sigma^2)$ distribution. Find the distribution of $Y = \frac{1}{\sigma^2} \sum_{i=1}^n\left(X_i-\mu \right)^2$. 
        \item Let $Y=X_1+X_2$, where $X_1$ and $X_2$ are independent, $X_1\sim\chi^2(\nu_1)$ and $Y\sim\chi^2(\nu_1+\nu_2)$, where $\nu_1$ and $\nu_2$ are both positive. Show $X_2\sim\chi^2(\nu_2)$. 
        \item Let $X_1, \ldots, X_n$ be random sample from a $N(\mu,\sigma^2)$ distribution. Show
\begin{displaymath}
    \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1),
\end{displaymath}
where $S^2 = \frac{\sum_{i=1}^n\left(X_i-\overline{X} \right)^2 }{n-1}$.
Hint: $\sum_{i=1}^n\left(X_i-\mu \right)^2 = \sum_{i=1}^n\left(X_i-\overline{X} + \overline{X} - \mu \right)^2 = \ldots$

For this question, you may use the independence of $\overline{X}$ and $S^2$ without proof. 
    \end{enumerate}

\end{enumerate}

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\noindent
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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/302f13} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f13}}

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