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\begin{center}
{\bf STAT 609 Second Exam}\\
{\bf 2:30pm-3:45pm, Nov. 20, 2015}
\end{center}

Please show all your work for full credits.
\begin{enumerate}
\item
Let $X$ and $Y$ be random variables having the joint pdf
\[ f(x,y) = \left\{ \begin{array}{ll}
x+y & \quad 0<x<1 , \ 0<y<1  \\
0 & \quad \mbox{otherwise} \end{array} \right. \]
\begin{enumerate}
	\item
	(5 points)
	Find ${\rm Cov} (X,Y)$.
	\item
	(5 points)
	Find $E(X|Y=y)$
	for $ 0<y<1$. 
	\item
	(5 points)
	Find $E(Y e^{ X+X^{-1}} |X=x )$ for $0<x<1$.
	\item
	(10 points)
	Find the mgf $M(t,s) = E(e^{tX+sY})$. (Hint: integration by parts) 
\end{enumerate}
\item
Suppose that the 2-dimensional random vector $(X,Y)$ is bivariate normal with pdf 
\[ f(x,y) = \frac{1}{2\pi\sqrt{1-\rho^2}} \exp \left\{ - \frac{x^2 -2\rho xy +y^2 }{2(1-\rho^2)}  \right\} \quad\quad (x,y) \in {\cal R}^2 \]
\begin{enumerate}
\item
(5 points)
Find the conditional pdf of $Y$ given $X$.
\item
(5 points)
Show that $X-Y$ and $X+Y$ are independent. 
\item
(10 points)
Let $U = \max \{ X,Y\}$ and $V= \min \{ X,Y\}$.
Find the joint pdf of $(V,U)$. (Hint: Show $P(U \leq u, V \leq v) = P(X \leq u, Y \leq u) - P(v \leq X \leq u, v \leq Y \leq u)$.)
\item
(10 points)
Find the pdf of $R= U-V$. 
\end{enumerate}
\item
Let $Z$ be a random variable having pdf
\[ f_Z(z) = \left\{ \begin{array}{ll}
e^{-z} &  \quad 0< z  \\
0 & \quad \mbox{otherwise} \end{array} \right.  \]
Let $X$ and $Y$ be random variables such that
given $Z= z$, $X$ and $Y$ are conditionally independent with conditional pdf's 
\[ f_X(x|z)= z e^{-zx} , \quad\quad
f_Y(y|z)= ze^{-zy}, \quad\quad x>0 , \ y>0  
\]
\begin{enumerate}
\item
(10 points)
Find the joint pdf of $(X,Y)$. 
(Hint: $\int_0^\infty t^{k-1} e^{-t}dt =\Gamma (k) = (k-1)!$)
\item
(10 points)
Calculate $E(Z| X=x, Y=y)$. 
\end{enumerate}



{\bf (One more problem on the 2nd page)}
\pagebreak 

\item
Let $(X_1,...,X_n)$ be a random sample from a population (the cdf of $X_1$) which has a pdf 
\[ f_\theta (x) = \frac{1-\theta^2}{2} \exp ( \theta x - |x|) , \quad\quad x \in {\cal R} \]
where $\theta $ is an unknown parameter satisfying $|\theta | <1$.  
\begin{enumerate}
	\item
(10 points)
	Find the joint pdf $f(x,y)$ of $(X_{(1)},X_{(n)})$, $x<y$, where $X_{(j)}$ is the $j$th order statistic.  
	Express $f(x,y)$ explicitly in three regions: $\{ 0 < x < y\}$, $\{ x < y < 0 \}$, and $\{ x < 0  < y\}$. 
\item
(5 points)
Using  Theorem 5.2.11, show that
$T = X_1+\cdots + X_n$ has a pdf from an exponential family. 
\item
(10 points)
For $n=2$, show (without using Theorem 5.2.11) that the pdf of $T=X_1+X_2$ is from an exponential family,
using the formula for the pdf of a sum of two random variables.  (Hint: you may leave an integral in the formula of the pdf and still argue that the pdf is from an exponential family.)
\end{enumerate}
\end{enumerate}
\end{document}
