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\begin{center}
{\bf STAT 609 Final Exam}\\
{\bf 2:30-4:30pm, Dec.\ 20, 2015}
\end{center}

Please show all your work for full credits.
\begin{enumerate}
\item
(8 points)
On the desk of a professor there are two boxes, $B_1$ and $B_2$, each containing 10 computer disks. Box $B_1$ contains 6 V disks and 4 C disks and box $B_2$ contains 2 V disks and 8 C disks. 
A box is selected with probabilities $P(B_1) = 3/4$ and $P(B_2 ) = 1/4$. A disk is then selected at random from the chosen box. 
\begin{enumerate}
	\item
	(5 points)
	Find the probability that the selected disk is a V disk.
	\item
	(5 points)
	Given that the selected disk is a V disk, find the probability that it was from box $B_1$. 
\end{enumerate}
\item
Suppose that the joint distribution of $(X,Y,Z)$ is multivariate normal with mean vector $(0,0,0)$ and covariance matrix
\[ \left( \begin{array}{ccc}
2 & 1 & 1 \\ 
1 & 3 & 0 \\
1 & 0 & 1 \end{array} \right) \]
\begin{enumerate}
\item
(5 points)
Calculate the covariance between $3X+2Y$ and $ X -3 Z$. Are they independent? 
\item
(5 points)
Calculate the covariance between $X-2Y+2Z$ and $ X +Y+ Z$. Are they independent? 
\item
(5 points)
Show that $\frac{(X+Y-Z)^2}{c} $ has the central chi-square distribution with \vspace{2mm} degree of freedom 1, and find the value of $c$. 
\item
(5 points)
Show that $\frac{(X+Y-Z)^2}{c} + \frac{(2X-Y+Z)^2}{d} $ has the central chi-square \vspace{2mm} distribution with degrees of freedom 2, and find the value of $d$. 
\item
(5 points)
Find the conditional distribution of $(X,Y)$ given $ Z=z$.  
\end{enumerate}
\item
Let the joint pdf of $(X,Y)$ be
\[ f(x,y) = \left\{  \begin{array}{ll}
x+y & \quad 0<x<1, \ 0<y<1 \\
0 & \quad \mbox{otherwise} \end{array}  \right. \] 
Let $U = X+Y$ and $V = X-Y$. 
\begin{enumerate}
\item
(5 points)
Show that the joint pdf of $(U,V)$ is
\[ g(u,v) = \left\{  \begin{array}{ll}
u/2 & \quad 0<u<1, \ -u <v< u \\
u/2 & \quad 1<u<2, \ u-2 <v< 2-u \\
0 & \quad \mbox{otherwise} \end{array}  \right. \]  
\item
(5 points)
Obtain the conditional pdf of $V$ given $U=u$. 
\item
(5 points)
For any given $t$, show that 
\[
E( e^{tV} | U=u) = \left\{ \begin{array}{ll}
\frac{e^{tu} - e^{-tu}}{2tu} & \quad 0<u<1 \vspace{3mm} \\
\frac{e^{t(2-u)} - e^{t(u-2)}}{2t(2-u)} & \quad 1<u<2 \\
\end{array} \right.
\]
{\bf (continued on the next page)}

\item
(5 points)
Use the result in the previous part to show that the mgf of $V$ is 
\[ M_V(t) = E(e^{tV}) = \frac{e^t+e^{-t}-2}{t^2}  \]
\end{enumerate}
\item
Let $X_n$ be a random variable having the chi-square distribution with degrees of freedom $n$, $n=1,2,...$
In the following, the limiting process is with respect to $n \rightarrow \infty$. 
\begin{enumerate}
\item
(6 points)
Show that 
\[ \frac{X_n}{\sqrt{2n}} - \sqrt{\frac{n}{2}} \ \ \mbox{converges in distribution to} \  N(0,1) \]
\item
(6 points)
For any positive integer $k$, show that 
\[ \frac{X_n^k}{\sqrt{2} n^{k-1/2}} - \sqrt{\frac{n}{2}} \ \ \mbox{converges in distribution to} \  N(0,k^2) \]
\item
(6 points)
Define 
\[ Y_n = \left\{ \begin{array}{ll}
0 & \quad X_n \leq 1 \\
X_n & \quad X_n > 1 \end{array} \right. \]
Show that $Y_n - X_n $ converges in probability to 0.
\end{enumerate}
\item
Let $X_1,...,X_n$ be a random sample with a pdf 
\[ f (x) = \left\{ \begin{array}{ll}
\theta \varphi^\theta x^{-(\theta + 1)}  & \quad x > \varphi   \\
0 & \quad x \leq \varphi \end{array} \right.  \]
where $\theta >0$ and $\varphi >0$ are fixed parameters.
\begin{enumerate}
\item
(5 points)
Find the pdf of the $j$th order statistic, $j=1,...,n$. 
\item
(7 points)
When $\varphi$ is known and $\theta $ is unknown, 
show that the family of pdf's indexed by $\theta$ is an exponential family and find a sufficient and complete statistic $T$ for $\theta$. 
\item
(7 points)
When $\theta$ is known and $\varphi $ is unknown, show that the minimum order statistic $X_{(1)}$ is
 a complete and sufficient statistic for $\varphi$.
\item
(8 points)
Suppose that $\varphi  = e^\theta $ is unknown. Show that the pair of statistics, $T$ and $X_{(1)}$, is minimal sufficient for $\theta$. 
\end{enumerate}
\end{enumerate}

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