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{\bf STAT 311 Old final exam problems}\\
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Dec 13. 2005
\begin{enumerate}




\item Suppose that $X_1,\cdots, X_n$ are a random sample of size
$n$ from Bernoulli distribution with $P(X_i=1)=p, 0<p<1$. (a)
Compute the correlation coefficient of $Y_1=X_1+X_2$ and
$Y_2=X_1+X_3$. Are $Y_1$ and $Y_2$ independent?
 (b) Determine the probability distribution of
 $X_1^2 +\cdots + X_n^2$.


\item A person is flipping a fair coin repeatedly until he obtains two heads.
Let $X$ be the number of tosses until he obtains two heads. (a)
Determine the probability mass function of $X$.
(b) Compute $\mathbb{E} X$\\
{\em Solution.} Note that $X \geq 2$. If $p$ is the probability of
obtaining head, $p(2)=p^2, p(3)={2 \choose 1}p^2(1-p), p(4)={3
\choose 1}p^2(1-p)^2, \cdots$. In general we have $p(x) = {x-1
\choose 1}p^2(1-p)^{x-2}$. $$\mathbb{E} X = \sum_{x=2}^{\infty}
x(x-1)p^2(1-p)^{x-2}= p^2\sum_{x=2}^{\infty} \frac{d^2}{dq^2}q^x,$$
where $q=1-p$.


\item $X$ and $Y$ are uniformly distributed over $0<x<1, x < y < x+1$. \\
(a) Compute the marginal densities of $X$ and $Y$.\\
(b) Are $X$ and $Y$ independent? Justify your answer.\\
(c) Compute $P(\frac{1}{2} <Y<\frac{3}{2})$.\\

\item The density of $X$ is given by $f(x) = |x| $ for $ -1 < x <1$ and $f(x)=0$ otherwise. \\
(a) Compute the cumulative distribution of $X$ and plot its graph.
(b) Determine the density function of $U=X^2$\\
\item Let $X_1$ and $X_2$ be independent Poisson random variables
with means
$\lambda_1$ and $\lambda_2$ respectively. Assume further that $\lambda_1 + \lambda_2=1$. Hint. The probability mass
function of a Poisson random variable with parameter $\lambda$ is given by
$p(x) = e^{-\lambda}\lambda^x/x!, x=0,1,2,\cdots$ \\
(a) Derive the moment generating function of $X_1$. \\
(b) Derive the probability mass function of $X_1 + X_2$. \\
(c) Compute $\mathbb{E}(X_1|X_1+X_2=z)$.\\

\end{enumerate}
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