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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
Suppose that the covariance matrix of the 2-dimensional random vector $(X,Y)$ is
\[ \left( \begin{array}{cc}
1 & 1 \\
1 & 4 \end{array} \right) \]
\begin{enumerate}
\item
(10 points)
Find the covariance matrix of $(X-2Y, 2X-Y)$.
\item
(10 points)
If the joint distribution of $(X,Y)$ is normal, show that it is from an exponential family with the form
\[ f_\theta (x,y) = h(x,y)c(\theta ) \exp \left( \sum_{i=1}^k w_i(\theta ) t_i(x,y)\right)  , \]
 and identify $k$, the value of the parameter vector $\theta $, and functions $h$, $c$,  and $t_i$'s.
\end{enumerate}
\item
Let $P$ be a random variable having pdf
\[ f(p) = \left\{ \begin{array}{ll}
3 p^2 &  \quad 0< p <1 \\
0 & \quad \mbox{otherwise} \end{array} \right.  \]
Suppose that given $P= p$, $X$ and $Y$ are conditionally independent discrete random variables with
\[ P(X=k|P= p)=P(Y = k|P= p ) = p(1-p)^k, \quad\quad k=0,1,2,... \]
\begin{enumerate}
\item
(10 points)
Obtain the pmf of $X$, i.e., $P(X=k)$, $k=0,1,2,...$
\item
(10 points)
Obtain the pmf of $Z = \max \{ X,Y\}$.
\item
(10 points)
Obtain the conditional pdf of $P$ given $X=x$ and $Y=y$, $x, y = 0,1,2,...$
\end{enumerate}
\item
Let $X$ and $Y$ be independent random variables, $X \sim uniform(0,1)$ and $Y \sim uniform(0,1)$.
\begin{enumerate}
\item
(10 points)
Obtain the joint pdf of $(X+Y,X-Y)$.
\item
(10 points)
Obtain the conditional pdf of $X| (X-Y) =t$.
\item
(10 points)
Obtain the pdf of $|X-Y|$.
\end{enumerate}
\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
(10 points)
Show that $Z \sim N(0,1)$.
\item
(5 points)
Prove that the joint distribution of $(X,Z)$ is not normal.
\end{enumerate}
\item
(5 points)
Let $X$ and $Y$ be two random variables having mgf's $M_X(t)$ and $M_Y(s)$ which are finite for $t \in (-h,h)$ and $s \in (-h, h)$ with a positive constant $h$.
Suppose that the mgf of the random vector $(X,Y)$ is
\[ M_{(X,Y)}(t,s) =E( e^{tX+sY})= M_X(t) M_Y(s) , \quad \quad t \in (-h,h), \  s \in (-h,h) \]
Show that $X$ and $Y$ are independent.
\end{enumerate}
\end{document}
