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\begin{center}
{\bf STAT 609 First Exam}\\
{\bf 2:45-4:45pm, Dec.\ 17, 2014}
\end{center}

Please show all your work for full credits.
\begin{enumerate}
\item
(8 points)
A person has probability 0.005 to have a type of disease. 
A medical test is positive with probability 0.98 
if a person has the disease, and is negative with probability 0.1 if a person does not have the disease. 
One person had a test and the result was positive. 
What is the probability that this person actually has the disease?
\item
Suppose that the joint distribution of $X$ and $Y$ is normal with covariance matrix
\[ \left( \begin{array}{cc}
2 & 1 \\ 
1 & 3 \end{array} \right) \]
\begin{enumerate}
\item
(6 points)
Find a value $a$ such that $X-aY$ and $2X+aY$ are independent. 
\item
(6 points)
Is the solution $a$ in the previous part unique? Why?
\end{enumerate}
\item
(8 points)
Let $U$ and $V$ be independent standard random variables, each has $N(0,1)$ distribution.
Find the distribution of $U/(U+V)$. 
\item
Let $X$ and $Y$ be independent and identically distributed random variables having the $exponential(1)$ pdf.   
\begin{enumerate}
\item
(6 points)
Obtain the joint pdf of $(X+Y,X-Y)$. 
\item
(6 points)
Obtain the conditional pdf of $(X+Y) | (X-Y)=t$. 
\item
(6 points)
Obtain the mgf of $X-Y$ and use the mgf to find $E(X-Y)^2$. 
\end{enumerate}
\item
Let $X_n$ be a random variable having the $Poisson(n)$ distribution, $n=1,2,...$
In the following, the limiting process is with respect to $n \rightarrow \infty$. 
\begin{enumerate}
\item
(7 points)
Show that 
\[ \frac{X_n}{\sqrt{n}} - \sqrt{n} \ \ \mbox{converges in distribution to} \  N(0,1) \]
\item
(6 points)
For any positive integer $k$, show that 
\[ \frac{X_n^k}{n^{k-1/2}} - \sqrt{n} \ \ \mbox{converges in distribution to} \  N(0,k^2) \]
\item
(7 points)
Define 
\[ Y_n = \left\{ \begin{array}{ll}
1 & \quad X_n = 0 \\
X_n & \quad X_n > 0 \end{array} \right. \]
Show that $Y_n - X_n $ converges in probability to 0.
\item
(6 points)
Show that the result in part (b) is still true when $k$ is a negative integer.  
\end{enumerate}
{\bf (Another problem is on the next page)}
\item
Let $X_1,...,X_n$ be a random sample with a pdf 
\[ f (x) = \left\{ \begin{array}{ll}
e^{- x^2/(2\sigma^2)}/c(\mu,\sigma)  & \quad x > \mu \\
0 & \quad x \leq \mu \end{array} \right.  \]
where $\sigma >0$ and $\mu \in {\cal R}$  and $c(\mu,\sigma) = \int_\mu^\infty e^{- t^2/(2\sigma^2)} dt$.
\begin{enumerate}
\item
(4 points)
Derive a formula of the pdf of the $j$th order statistic, $1 \leq j \leq n$. Do not try to simplify integrals of the form 
$\int_\mu^x e^{- t^2/(2\sigma^2)} dt$.  
\item
(8 points)
In the case where $\mu$ is known and $\sigma $ is unknown, obtain a complete and sufficient statistic for $\sigma$.  
\item
(8 points)
In the case where $\mu$ is unknown and $\sigma $ is known, obtain a complete and sufficient statistic for $\mu$.
\item
(8 points)
Suppose that $\sigma^2 = \mu^2$ is unknown. Obtain a minimal sufficient statistic. 
\end{enumerate}
\end{enumerate}

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