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\begin{center}
{\bf STAT 609 First Exam}\\
{\bf 11:00am-12:00noon, Oct.\ 3, 2014}
\end{center}

Please show all your work for full credits.
\begin{enumerate}
\item
Let $A$, $B$, $C$, and $D$ be four events. Suppose that if all $B$, $C$,  and $D$ occur, then $A$ occurs.
Show that $P(A) \geq P(B) + P(C) +P(D) -2$.
\item
A red die and a white die are rolled, where each die has 6 faces numbered $1,...,6$.
\begin{enumerate}
\item
 Are the events $A = \{\mbox{4 on the red die} \}$ and $B= \{ \mbox{sum of dice is odd} \}$ independent?
\item
 Are the events $C = \{\mbox{5 on the red die} \}$ and $D= \{ \mbox{sum of dice is 11} \}$  independent?
 \item
 Calculate the conditional probability that the sum of dice is larger or equal to 11, given that 5 on the red die.
\end{enumerate}
\item
Let $X$ be a random variable having pdf
\[ f (x) = \left\{ \begin{array}{ll}
x & \quad 0<x \leq 1 \\
2-x & \quad 1 < x \leq 2 \\
0 & \quad \mbox{otherwise} \end{array} \right.  \]
\begin{enumerate}
\item
Find the cdf of $X$.
\item
Obtain $E(X)$.
\item
If $Y = X^2$, find the pdf of $Y$.
\item
Find $E(Y)$, and use the result to find Var$(X)$.
\end{enumerate}
\item
Let $X$ be a discrete random variable with $P(X=a)=p $ and $P(X=b)=1-p$,
where $0<p<1$ and $0<a<b$.
Show that
\[ \lim_{n\rightarrow \infty} \frac{E(X^{n+1})}{E(X^n)} = b \]
\item
Let $X$ be a discrete random variable with pmf
\[ f_p(x) = \frac{1}{\log (1/p)} x^{-1} (1-p)^x, \quad \quad x=1,2,... \]
where $0<p<1$.
\begin{enumerate}
\item
Obtain $E(X)$ by directly calculating
\[ E(X) = \sum_{x=1}^\infty x f_p(x) \]
\item
Using the fact that $f_p(x)$ is a pmf for any $p \in (0,1)$, show that the mgf of $X$ is
\[ M_X(t) = \frac{ \log (1-(1-p)e^t)}{\log (p)} , \quad \quad t < - \log (1-p) \]
\item
Find $E(X)$ by differentiating $M_X(t)$ and show the result is the same as that in part (a).
\end{enumerate}

\end{enumerate}

\end{document}
