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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam 
(Part I of Written Exam)\\
Sunday,  January 7, 2001
\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 We draw 52 cards without replacement from a well-shuffled
deck of 52 cards. We place them in a line from left to right. We
say there is a Jack-Queen contact in positions $(i,i+1)$ if
position $i$ has a Jack and position $i+1$ has a Queen or if
position $i$ has a Queen and position $i+1$ has a Jack.
\begin{parts}
\PP What is the expected number of Jack-Queen contacts?

\PP Use Markov's inequality to give an upper bound for the
probability that there is at least one Jack-Queen contact.
  
\end{parts}



\QQ Consider the simple regression model
\[Y_i = \beta_0 +  \beta_1 X_i +\epsilon_i\]
with Var($\epsilon_i$) = $\sigma^2$.
The  least squares estimates for $\beta_0$ and $\beta_1$ are 1.84 and
1.09,
respectively. The estimated variances for $\hat{\beta}_0$ and
$\hat{\beta}_1$
are
\begin{eqnarray*}
s^2(\hat{\beta}_0) =
\hat{\sigma}^2\left(\frac{1}{n}+\frac{\bar{x}^2}{SXX}\right)=0.124, & &
 s^2(\hat{\beta}_1) = \hat{\sigma}^2/SXX = 0.025,
\end{eqnarray*}
where $SXX$ is the corrected sum of squares for $X$,
$\bar{x}$ is a positive number, $\hat{\sigma}^2=1.64$,  and
$n=35$.
\begin{parts}
\PP
Based on the information given above,  estimate Cov($\hat{\beta}_0$,
$\hat{\beta}_1$).

\PP Let $\hat{Y}$ be the fitted value at $X=4$.  Estimate the
variance of $\hat{Y}$ if $\hat{Y}$ is being used to estimate the
mean value of $Y$ at $X=4$.
\end{parts}




\QQ Suppose that when measuring the {\bf radius} of a circle, an
error is made that has a $N(0,\sigma^2)$ distribution with
$\sigma^2$ unknown.  If $n$ independent measurements are made,
find the best unbiased estimator of the {\bf area} of the circle
(and prove it is best unbiased).


\QQ A sociologist is examining the personalities of students in
different major fields.  She has randomly selected ten students
from each of five majors, and plans to do an analysis of variance
on each of the subscores of the personality test.  The five
fields are Math, Computer Science, English, History, and Art.

Suppose she gets the ANOVA results and the overall F test from
the first subscore, a measure of intraversion/extraversion, is
significant.  (An introvert is primarily concerned with his/her
own mental life, not the lives of others.)  Assuming the usual
ANOVA assumptions are satisfied, what further tests would you do
to examine the differences among the five majors? Justify your
choice.



\QQ A certain community is composed of $m$ families, $n_i$ of
which have $i$ children, $\sum_{i=1}^r n_i = m$\, ($r$ is the
number of children in the largest family).  If one of the
families is randomly chosen, let $X$ denote the number of
children in that family. If one of the children is randomly
chosen, let $Y$ denote the total number of children in the family
of that child. Show $E[Y] \geq E[X]$.


\QQ Consider a sample of $n=68$ randomly selected individual
  tax returns for 1999 prepared by a local branch of a national
  tax preparation service. The following multiple linear
  regression models are run :
$$ Y=\beta_0+\beta_1X_1
+\beta_2X_2+\beta_3X_3+\beta_4X_4+\beta_5X_1X_4+\beta_6X_2X_4+
\beta_7X_3X_4 +\epsilon $$
$$
Y=\beta_0+\beta_1X_1 +\beta_2X_2+\beta_3X_3+\beta_4X_4
+\epsilon $$
with the sums of squared residuals equal 401.61 and
504.88, respectively, where $X_1$ represents total income; $X_2$
is earned income; $X_3$ is federal itemized or standard
deductions; $X_4$ represents marital status ($X_4=1$ if married,
$X_4=0$ if single); $Y$ represents total tax paid as a percent of
total income. Using a 5\% level of significance, test the
hypothesis that $\beta_5=\beta_6=\beta_7=0$ versus the
alternative that at least one of these beta coefficients is not
zero.



\QQ Suppose that $X_n \sim AsN(\theta, \sigma_n^2(\theta))$ (see
definition below), where $\theta$ is a real-valued parameter.  A
differentiable function $g$ is called a variance stabilizing
transformation for $X_n$ if $g(X_n) \sim AsN(g(\theta), \xi^2_n)$,
where $\xi_n$ does not depend on $\theta$.

\begin{parts}
  
\PP Assume that $\sigma_n(\theta) = h(\theta) \cdot \nu_n$, where
$\nu_n \rightarrow 0$.  Show that a variance stabilizing
transformation satisfies the differential equation $g'(x) =
c/h(x)$, where $c$ is a constant.

\PP Find \underline{\bf a} variance stabilizing transformation
for $X_n$, where $X_n = Y_n/n$ and $Y_n \sim
\mbox{Binomial}(n,p)$.

\item[Definition:] $AsN$ stands for ``asymptotically normal''.
  To be precise, $X_n \sim AsN( \mu, \sigma^2_n)$ means that the
  sequence $\{ X_n \}$ satisfies \[\frac{X_n - \mu}{\sigma_n}
  \stackrel{D}{\longrightarrow} N(0,1) \;\; \mbox{ as $n
    \rightarrow \infty$}\,.\]

\end{parts}


\QQ Let $X_1, X_2, X_3, \ldots$ be i.i.d.\ random variables.  Let
$N$ be a nonnegative integer-valued random variable which is
independent of the $X_i$'s.  Show that
\[ \mbox{Var}\left(\sum_{i=1}^N X_i \right) = 
E(N) \mbox{Var}(X_1) + (EX_1)^2 \mbox{Var}(N)\,.\]
(Assume that both $EN^2$ and $EX_1^2$ are finite.)


\QQ Let $X_1, X_2, \ldots, X_n$ be a random sample from a uniform
distribution on the interval $(\theta, 1/\theta)$ where $0 <
\theta < 1$.  Find the maximum likelihood estimator of $\theta$.



\end{document}
