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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam 
(Part I of Written Exam)\\
Thursday, January 2, 1997
\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.}


\vsp


\QQ  
%Hollander -- M.S. Math.  Statistics
Let $X$  be a non-negative random variable with density function $f(x)$ and distribution function
$F(x)=P(X\leq x)$.  The survival function $S(x)$ is $S(x)=1-F(x)=P(X>x)$.
The failure rate function is
\[\lambda(x) =f(x)/S(x).\]
Show that $S(x)=\exp\{-\int_0^x\lambda(u)du\}$.


\QQ
%Leysieffer -- M.S. Probability
Let $\{X_1, X_2, \ldots, X_n\}$ be mutually independent random variables with
\[ P(X_k\leq t) =\left\{ \begin{array}{lll}
0,     &  \mbox{if  $t<0$},      &  \\
t^k,   &  \mbox{if $0\leq t <1$}, & \hspace{0.2in} k=1,2,\ldots, n \\
1,     & \mbox{if $1\leq t$ }     &
\end{array} \right. \]

\begin{parts}
\PP
Let $Y=\max(X_1, X_2, \ldots, X_n)$. Find E$Y$.

\PP
Let $Z=\prod_{i=1}^n X_i$. Find E$Z$.
\end{parts}

\vsp

\QQ
%Song--- MS Probability
While rolling a balanced die successively, the first 6 occurred on
the third roll. Conditional on this, what is the expected number of
rolls until the first 1 appears?

\newpage
\QQ
%Lin--MS Mathematical Statistics
Let $\{X_1, X_2, X_3, X_4, X_5\}$ be a random sample from an exponential distribution
having density function
\[f(x|\theta) = \left\{ \begin{array}{ll}
\theta\exp\{-\theta x\}, & \mbox{for $x>0$, $\theta>0$}, \\
0,                     &  \mbox{Otherwise}.
\end{array}\right.\]
Define the order statistics $X_{(1)}\leq X_{(2)}\leq X_{(3)}\leq  X_{(4)}\leq X_{(5)}$.

\begin{parts}
\PP
Derive the density function of  $X_{(3)}$, the median.

\PP
Derive E$(X_{(3)})$.
\end{parts}


\QQ
%Huang--MS  Math. Statistics
Suppose $X_1, \dots, X_n$ are i.i.d.  and have a uniform distribution
on the interval $[0, \theta]$, $\theta>0$.
\begin{parts}
\PP
Find the probability density function of $\max_i X_i$.
\PP
Show that $n(\theta-\max_i X_i)$ converges in distribution
to an exponential random variable with mean $\theta$.
\end{parts}


\QQ
%Lin--MS Statistics
Let $X_1, \ldots, X_n$ be independent and identically distributed
Bernoulli random variables with parameter $\theta\in (0,1)$.

\begin{parts}
\PP
Show that $s^2 = \frac{1}{n-1}\sum_{i=1}^n(X_i-\bar{X})^2$ is an unbiased 
estimation of $\mbox{Var}(X_1)$ for $n\geq 2$.

\PP
Show that, for $n=1$, there does not exist any unbiased estimator for 
$\mbox{Var}(X_1)$.
\end{parts}


\QQ
%Zahn--Applied Statistics
An attorney has called you, saying:

\vspace{0.1in}

My client is contesting the results of last November's election which he
lost by a narrow margin to the incumbent.  In this election contest, 27,000
absentee ballots were cast.  The official count showed that 16,132 were cast
for the incumbent.

\vspace{0.1in}

However, not all the absentee voters used a No. 2 pencil to darken the
circles on the mark sense form on which they recorded their vote.  The
Supervisor of Elections went through the ballots and darkened with pencil
those that appeared to be filled in with pen.

\vspace{0.1in}

After the election results were announced, my assistant and I went through
the 27,000 absentee ballots and identified all those that appeared to have
been darkened by the Supervisor of Elections.  We found a sample of 6498
such ballots, 3996 of which were marked for the incumbent.  Is this tendency
for more of the darkened ballots to be for the incumbent surprising?

\vspace{0.1in}

How would you answer the attorney's question?


\newpage
\QQ
%meeter -- Applied Statistics
In the standard regression model
\begin{eqnarray*}
Y & = &  X\vecc{\beta}+\vecc{\epsilon},
\end{eqnarray*}
where  $X$ is an $n\times p$ matrix with rank $p$,
 the first column of $X$ is $(1,1, \ldots, 1)'$,
and $\mbox{Cov}(\epsilon)=\sigma^2 I_n$.

\begin{parts}
\PP
Let $\hat{\vecc{\beta}}$ be the least squares estimate of $\vecc{\beta}$
and $\hat{Y}=X\hat{\vecc{\beta}}$.
Show  that $\sum_{i=1}^n\mbox{Var}(\hat{y}_i) = p\sigma^2$.

\PP
Show that $\mbox{Var}(\sum_{i=1}^n \hat{y}_i) = n\sigma^2$.
\end{parts}

\QQ
%Zahn--Applied Statistics
 A two-way factorial experiment has been done.  Factor A has three
levels,  and Factor B has four levels. There are two replications,
that is, there are two observations in each of the 12 cells.  Factor A
is a fixed effect and Factor B is a random effect.
Denote the variance of the experiment error by $\sigma^2$,
the variance of  the random effect for Factor B by $\sigma^{2}_b$,
and the variance of the interaction effect by $\sigma^2_{ab}$.
The expected mean squares are E$[MSA]=\sigma^2+2\sigma^2_{ab}+c$,
E$[MSB]=\sigma^2+6\sigma^2_b$, E$[MS(AB)]=\sigma^2+2\sigma^2_{ab}$,
and E$[MSE]=\sigma^2$.


\begin{parts}
\PP
 Construct the appropriate ANOVA table, giving the sources of variation,
degrees of freedom.

\PP
How would you test the hypothesis that Factor A has no effect?  Factor
B?  Why are the denominators of the test statistics for these two tests
different?

\PP
Suppose you did the F-statistic for the significance of Factor A using a
computer package and that the output said  ``p-value = .0003. "  Interpret
this output.

\PP
Suppose you did the F-statistic for the significance of Factor B using a
computer package and that the output said  ``p-value = .0000. "  Interpret
this output.
\end{parts}


\end{document}


