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\begin{center}
{\bf STAT 609 Second Exam}\\
{\bf 11:00am-12:00noon, Nov. 7, 2014}
\end{center}

Please show all your work for full credits.
\begin{enumerate}
\item
Let $X$ and $Y$ be independent discrete random variables with 
\[ P(X=k)=P(Y = k) = p(1-p)^k, \quad\quad k=0,1,2,... \]
where $ p \in (0,1)$ is a fixed constant. 
Obtain the pmf of $Z = \max \{ X,Y\}$. 
\item

\item
Suppose that the covariance matrix of the 2-dimensional random vector $(X,Y)$ is 
\[ \left( \begin{array}{cc}
1 & 1 \\
1 & 4 \end{array} \right) \]
Find the covariance matrix of $(X-2Y, 2X-Y)$. 
\item
Let $X$ and $Y$ be independent random variables, $X \sim uniform(0,1)$ and $Y \sim uniform(0,1)$. 
Obtain the pdf of $|X-Y|$. 
\item
Let $X$ and $Y$ be independent random variables, $X \sim N(0,1)$, $Y \sim N(0,1)$, and 
\[ Z = \left\{ \begin{array}{rl}
|Y| & \quad X \geq 0 \\
-|Y| & \quad X < 0 \end{array} \right. \]
\begin{enumerate}
\item
Show that $Z \sim N(0,1)$. 
\item
Show that the joint pdf of $(X,Z)$ is not normal. 
\end{enumerate}
\end{enumerate}

\end{document}
