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\begin{center}
{\bf STAT 609 First Exam}\\
{\bf 1:00pm-2:15pm, Oct.\ 7, 2015}
\end{center}

Please show all your work for full credits.
\begin{enumerate}
\item
(10 points)
Let $A$ and $B$ be events with $0<P(A)<1$ and $0<P(B)<1$. Suppose that $P(A|B) >P(A|B^c)$, where the superscript $c$ denotes complement.
Show that $P(B|A) > P(B)$. 
\item
(20 points)
A urn contains six balls represented by $1,...,6$. One ball is to be drawn at random from the urn. Let 
$A = \{ 1,2 \}$, $B = \{ 1,3\}$, $C= \{ 1,2,3,4\}$, and $D = \{ 1,2,5 \}$ be events. 
\begin{enumerate}
\item
 Are the events $A $ and $B$ independent?
\item
Are the events $A $ and $B$ conditionally independent given $C$?
\item
Are the events $D $ and $B$ independent?
\item
Are the events $D $ and $B$ conditionally independent given $C$?
\end{enumerate}
\item
(30 points)
Let $X$ be a random variable having pdf
\[ f (x) = \left\{ \begin{array}{ll}
a x^2 & \quad 0< x \leq 1 \\
\frac{b}{x^2}  & \quad 1 < x \leq 2 \\
0 & \quad \mbox{otherwise} \end{array} \right.  \]
where $a>0$ and $b>0$ are constants.
\begin{enumerate}
	\item
	Find the value of $a$ and $b$ such that $E(X^2)=1$. 
\item
Find the cdf of $X$.
\item
Find the pdf of  $Y = (X-1)^2$.
\end{enumerate}
\item
(15 points)
Let $x_1,...,x_n$ be $n$ numbers satisfying $\sum_{i=1}^n x_i =0$. Prove the following inequalities using the three well-known inequalities in probability theory. 
\begin{eqnarray*}
{\rm (a)} & & \left(\frac{1}{n}\sum_{i=1}^n |x_i| \right)^2 \leq \frac{1}{n} \sum_{i=1}^n x^2_i \vspace{2mm}  \\
{\rm (b)} & & \left(\prod_{i=1}^n |x_i| \right)^{1/n} \leq \frac{1}{n} \sum_{i=1}^n |x_i| \\
{\rm (c)} & & \frac{\mbox{ number of $|x_i|$'s $> \epsilon$}}{n} \leq \frac{1}{\epsilon^2 n} \sum_{i=1}^n x_i^2 \quad \mbox{for any $\epsilon >0$} 
\end{eqnarray*} 
Hint: Construct a random variable with certain distribution and then apply some inequalities.
\\


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

\item
(25 points)
Let $X$ be a discrete random variable with pmf
\[ f_\theta (x) = \frac{\gamma (x) \theta^x}{\phi (\theta )}, \quad \quad x=0,1,2,... \]
where $\theta > 0$ is a fixed constant, $\gamma (x) \geq 0$ is a function of $x$, and 
\[ \phi (\theta ) = \sum_{x=0}^\infty \gamma (x) \theta^x  \]
is assumed to be finite for any $\theta >0$. 
\begin{enumerate}
	\item
	Show that the family of pmf's, $\{ f_\theta , \theta >0 \}$, is an exponential family. 
\item
Obtain the moment generating function $M_X(t)$ in terms of a function of $t$ and $\theta$. 
\item
Obtain $E(X)$ by differentiating the moment generating function. 
\item
Show that $E(X)$ can also be obtained by differentiating $\phi (\theta )$ and interchanging the differentiation and 
summation (you need to justify why they can be interchanged). 
\item
Suppose that $X_1,...,X_n$ are independent and identically distributed with $f_\theta (x)$ and $S= \sum_{i=1}^n X_i$. 
Using the moment generating function, show that the pmf of $S$ is 
\[ g_\theta (s) = \frac{\gamma_n (s) \theta^s}{[\phi (\theta )]^n}, \quad \quad s=0,1,2,... \]
where  $\gamma_n (s) \geq 0$ is a function of $s$ defined by 
\[ [\phi (\theta )]^n = \sum_{s=0}^\infty \gamma_n (s) \theta^s  \]
 
\end{enumerate}

\end{enumerate}

\end{document}
