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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam 
(Part I of Written Exam)\\
Monday, January 30, 1995
\end{center}


\vsp
You have three hours.  Do as many problems as you can.  No one is
expected to answer all the problems correctly.  Partial credit will be
given.  All problems are worth an equal amount of credit.

\vsp
{\bf Put your solution to each problem on a separate sheet of paper.}


\QQ
Consider a data set containing 300 observations on $Y, X_1, X_2, \ldots,
X_{40}\,.$  The $X$'s are explanatory variables to be used in explaining
variation in $Y$.

\begin{parts}

\PP
Describe a procedure for identifying good multiple regression equations
here.  Include in your description comments on the criterion or criteria
you are using, what computer software you would use at various stages in
this process, and how you would choose among the millions of potential
regression equations here.  

\PP
Discuss briefly the consequences of having
too few or too many variables in the final model you recommend.

\end{parts}


\QQ A competition is held between two boats, $A$ and $B$.  The
competition involves two rounds.  The first round consists of six races.
If one of the boats has won more races than the other at the end of the
first round, then that boat is declared winner of the competition.
Otherwise a second round is held in which races are run until one of the
boats has won two {\it consecutive} races.  Suppose that races are
independent of one another and boat $A$ wins a race with probability
1/2, boat $B$ wins with probability 1/2.  Let $X$ denote the number of
races in the competition.  Find the probability mass function of $X$.




\QQ
Tom Bayes has purchased a coin from a novelty shop
and wishes to estimate the probability $p$ of
getting a head when this coin is tossed.  Tom suspects the
coin is biased, but knows nothing about the direction of the bias.
He decides to use a mixture of Beta distributions as his prior
distribution for $p$,
with density
\( \pi(p) = 2(p^3 + (1-p)^3),\, 0<p<1. \)
He then tosses the coin 4 times and observes 2 heads and 2 tails.
Find Tom's posterior distribution for $p$ and compute the mean 
and variance of this posterior distribution.

\pagebreak

\QQ
Let $X_1, X_2, \ldots, X_n$ be a random sample from a distribution
with density 
\[f(x|\theta) = \frac{\theta}{(1+x)^{\theta+1}}
\;\; \mbox{ for } 0 < x < \infty,\, \theta>0\,. \]

\begin{parts}

\PP Find the MLE $\hat{\theta}$ of $\theta$.

\PP Show that $\hat{\theta}$ is a complete sufficient statistic for
$\theta$.

\PP Is  $\hat{\theta}$ unbiased?

\end{parts}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% submitted by Niu.
% Applied Statistics


\QQ
(An $F_{\alpha=0.05}$ table is needed for this problem)
A pharmaceutical company would like to examine the potency
of a liquid medication mixed in large vats. To do this, a random sample
of three vats from a month's production  was obtained and five separate 
samples were selected from each vat. The results are listed in the following
table:

\vspace{0.1in}

\begin{tabular}{c|ccccc} \hline
Vat 1 &  3.2 & 3.8 & 3.5 & 3.0 & 3.4  \\
Vat 2 &  2.6 & 2.9 & 2.8 & 2.0 & 2.5  \\
Vat 3 &  3.4 & 3.9 & 3.3 & 3.1 & 3.6  \\\hline
\end{tabular}

\vspace{0.1in}

A random-effects model for this experiment has the form:
\begin{eqnarray*}
Y_{ij} & = & \mu +v_i + \epsilon_{ij}, \hspace{0.2in} i=1,2,3; \hspace{0.1in} j=1, \ldots, 5,
\end{eqnarray*}
where $\mu$ is the overall mean, the $v_i$'s are iid $N(0, \sigma^2_{v})$
random variables and the $\epsilon_{ij}$'s are iid $N(0, \sigma^2_{\epsilon})$ variables.
Furthermore, $\{v_i\}$ and $\{\epsilon_{ij}\}$ are assumed to be independent.

\begin{parts}

\PP
Given  $\bar{Y}_{1\cdot}=3.38$, $\bar{Y}_{2\cdot}=2.56$,
$\bar{Y}_{3\cdot}=3.46$ and $\bar{Y}_{\cdot\cdot}=3.13$.
Test the null hypothesis $H_0: \hspace{0.05in} \sigma_{v}^2=0$ vs
the alternative hypothesis $H_a: \hspace{0.05in} \sigma_{v}^2>0$ by using
 $\alpha=0.05$.

\PP
Let $MSA$  be  the mean squares among groups and $MSW$ the mean squares
within groups. Show that $\hat{\sigma}^2_{\epsilon} = MSW$ and 
 $\hat{\sigma}^2_{v} = (MSA-MSW)/5$ are unbiased estimates for $\sigma^2_{\epsilon}$
and $\sigma^2_{v}$, respectively.

\end{parts}


\QQ Let $X$ and $Y$ be i.i.d.\ integer
valued random variables.  Show that
\[P(X + Y\ \mbox{is even})\geq 1/2.\]


\QQ Let $X$ and $Y$ be nonnegative random variables such that $ XY \ge
1$ (a.s.)  Show that $ EX \cdot EY \ge 1$.



\end{document}

