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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam\\
Friday,  August 23, 2002
\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.}

\vsph



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

\QQ Answer the following.  The parts are unrelated; the solution
to (b) does not use (a).

\begin{parts}
  \PP Let $(X,Y)$ have the joint pdf
  \begin{displaymath}
    f(x,y)= \left\{
        \begin{array}{ll}
2 \cdot \frac{1}{2\pi} e^{-(x^2+y^2)/2} & \mbox{if $xy>0$}\\
0 & \mbox{otherwise}
        \end{array}
\right.
  \end{displaymath}
Find the marginal distributions of $X$ and $Y$.

\PP  Let $S \sim N(0,1)$.  Observe $S$; then toss a fair coin and
define $T$ as follows:
  \begin{displaymath}
    T= \left\{
        \begin{array}{rl}
S & \mbox{if the toss is ``heads'',}\\
-S & \mbox{if the toss is ``tails''.}
        \end{array}
\right.
  \end{displaymath}
  Show that $T$ has a normal distribution, but that $S+T$ does
  not.

\end{parts}


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

\QQ A box contains tickets numbered 1 to $N$. Let $X$ denote the largest
number drawn in $m$ random drawings with replacement.

\begin{parts}

\PP Find $P(X \leq k)$.

\PP Find $P(X = k)$.
  
\end{parts}


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

\QQ Let $X_1, X_2, \dots, X_n, \dots $ be independent and
identically distributed {\bf continuous} random variables.
\begin{parts}
  \PP Determine the values of $P\{X_1 \geq
  X_2\}$ and $P\{X_1 \geq X_2 \geq X_3\}$.  Also find the value
  of $P\{X_1 \geq X_2 \geq \cdots \geq X_m\}$ for arbitrary $m$.

\PP Define a discrete random variable $N$ such that
$$
X_1 \geq X_2 \geq \dots \geq X_{N-1} < X_N
\ .
$$
That is, $N$ is the point at which the sequence stops
decreasing.  Use your answer to part (a) to show that $E[N] = e$.
  
\end{parts}



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

\QQ Consider the ``regression through the origin'' model: $Y_i=
\beta X_i +\epsilon_i, i=1, 2, ..., n$, where the $\epsilon_i$
are i.i.d. $N(0,\sigma^2)$.

\begin{parts}
  
  \PP Find the least squares estimate $\hat \beta$ of $\beta$ and
  show it is unbiased.
  
  \PP Obtain the standard error of $\hat \beta$.

\end{parts}

\QQ
Once upon a time in an age of interplanetary 
travel and trade, the drug Sarpie became a very important 
export for the planet of Kibtee. Sarpie was the only known 
drug which could cure a certain fatal bacterial infection. 
Sarpie was extracted from the stinger of an insect called 
bartuck. Like bees on earth, bartucks lived in hives which 
were populated by one queen bartuck. There were two different 
methods of keeping the hives and three different strains of 
bartucks. The following table shows the total Sarpie production
for three different hives under the different categories of 
strains and methods. 

\begin{table}[ht]
\begin{center}
\caption{ }
{
\begin{tabular}{r|ccccc|rc} \hline 
\multicolumn{1}{c|}{} & Strain A $\quad$ & Strain B
  $\quad$ & Strain C $\quad$ & $\quad $ Row Ave. $\quad $   
 \\ \hline\hline
Method 1   &51, 54, 48  & 55, 54, 59   & 42, 43, 44  & 
50\\ \hline
Method 2  & 48, 49, 53  & 53, 49, 54 & 43, 39, 38 &  47.33 \\\hline
Column Ave.  & 50.5  &  54 &  41.5  &     \\\hline
 
\end{tabular} }
\end{center} 
\end{table}

\begin{parts}
\PP Construct the ANOVA table and test to see if interaction 
effects between methods and strains are present 
at $\alpha=0.01$ level.
 
%\PP Test whether or not method main effects are present 
%at $\alpha=0.01$ level. 

\PP Test whether or not strain main effects are present 
at $\alpha=0.01$ level. 
  
\end{parts}

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

\QQ Statistics are widely used and abused.  Briefly analyze the
following situations.  (In each of the questions below, a
carefully chosen graph could be used to make your point.)

\begin{parts}
  
  \PP A sample of 1000 observations from a process {\bf over
    time} yields a mean of 18.5, with a minimum of 14.0 and a
  maximum of 24.7.

\begin{clist}
\item[1.] Some will contend that 
the best prediction of the next observation of this process is 18.5.
         Under what conditions would you agree with them?
\item[2.] What would lead you to predict 14.0?
\item[3.] What would lead you to predict 24.7?  
\end{clist}


\PP Two variables have been measured of 300 people.  For this
sample, $r = .04$, with a $p$-value of $.29$.  Some will
contend that this is compelling evidence to argue that the two
variables are not associated.
\begin{clist}
\item[1.] Under what conditions would you agree with them?
\item[2.] Under what conditions would you disagree with them?
\end{clist}


\PP $X$ and $Y$ are measured on 470 people.  $Y$ is regressed on
$X$, producing the equation $\hat{Y} = 3.09 + .00037X$, and the
$p$-value associated with .00037 is said to be .004.  Is it
possible that such a small regression coefficient could be
significant or has an error been made somewhere?  Justify your
answer.
  
\end{parts}

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

\QQ Consider the family of pdf's 
\begin{displaymath}
f(x) =\frac{\theta-1}{x^{\theta}}\;\;\;
\mbox{for $x \geq 1$\,\,\, (and $f(x) = 0$ otherwise)}
\end{displaymath}
where $\theta>1$.

\begin{parts}
  \PP Compute the Fisher information $I(\theta)$ for this family.
  \PP Given a sample of size $n$, find the Cram\'er-Rao lower
  bound for the variance of an unbiased estimate of $\theta$.
\end{parts}

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

\QQ We define $X \sim \mbox{Pois}(\lambda)$ to mean that $X$ has
a Poisson distribution with pmf $f(x) = \lambda^x
e^{-\lambda}/x!$ for $x = 0, 1, 2, \ldots$.  Suppose $X_1, X_2,
\ldots, X_n$ are independent random variables with $X_i \sim
\mbox{Pois}(e^{\alpha + \beta i})$.  Find a complete sufficient
statistic for $(\alpha,\beta)$.


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

\QQ Let $X_1, X_2$, and $X_3$ be a sample of size three from a
population having probability density function $f(x)=\lambda
e^{-\lambda x}$ for $x>0$ (and $f(x)=0$ otherwise) where
$\lambda>0$.


\begin{parts}
  \PP Find the maximum likelihood estimator (MLE) for the median
  of the distribution.

\PP
Show that the MLE is unbiased.

%\PP Identify the sample median.

\PP
Show that the sample median is a biased estimate of the median.

\PP Using an appropriate measure, decide whether the MLE or the
sample median is a better estimate for the population median.
Justify your answer.

\end{parts}


\end{document}

