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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam 
(Part I of Written Exam)\\
Wednesday, January 3, 1996
\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  
%Xufeng Niu -- Applied Stat
(Note: An $F_{\alpha=0.05}$ table is provided for this problem.)
In a complete factorial experiment, consider the following  mixed-effects model:
\begin{eqnarray*}
Y_{ij} & = & \mu +\alpha_i + \beta_j + \epsilon_{ij}, \hspace{0.2in} i=1,\ldots, a; 
\hspace{0.1in} j=1, \ldots, b,
\end{eqnarray*}
where  $\mu$ is the overall mean, $\alpha_i$ is a fixed effect corresponding to the $i$th
level of factor $A$, and $\beta_j$ is a random effect due to the $j$th level of factor $B$.
The  $\beta_j$'s are iid $N(0, \sigma^2_{\beta})$ variables  and the $\epsilon_{ij}$'s are iid
 $N(0, \sigma^2_{\epsilon})$ variables.
Furthermore, $\{\beta_j\}$ and $\{\epsilon_{ij}\}$ are assumed to be independent. 
 Let $MSA$ and $MSB$ denote the mean squares for factors $A$ and $B$,
respectively, and let $MSE$ be the mean square for error.

\begin{description}
\item[\bf (a)]    Calculate the expected value of $MSA$. 

\item[\bf (b)]  Given  $a=3, b=10, MSA=5.50, MSB=7.25$ and $MSE=0.62$,
perform an analysis of variance for the experiment using  $\alpha=0.05$.
 State your null and 
alternative hypotheses and summarize your results in a table.
\end{description}

\QQ
%Kaisheng Song -- Prob
A box contains three balls: one white, one
red, and one blue. Balls are successively and
randomly drawn from the box, with replacement after each draw. 
Let $N$ be the number of draws 
required for each color to appear at least once.  Find $P(N=n)$, where $n\geq
3$. (Hint: consider $P(N\geq n)$)
 

\QQ
%Meeter -- Applied Stat
Suppose you want to test three brands of sunscreen (sun-blocking cream).\\

Design 1 is to select thirty people at random from the target population and apply Brand A to the backs of ten people, B to another ten, and C to another ten, selection of subjects at random. Subjects are given the same number of hours under a tanning lamp and the amount of sunburn is measured.\\

Design 2 is to select ten people at random from the target population and apply Brands A, B, and C in side-by-side vertical stripes on the back of each person.  The response measured in the same way as Design 1.\\

\begin{parts}
\PP
Give the models and ANOVA tables for designs 1 and 2.  Indicate Source, df, and Expected Mean Squares.

\PP
How would you select the positions for A, B, and C in Design 2?  Explain the advantages of your method.


\PP
Which design would you prefer to use? Explain the reasoning for your choice.
\end{parts}

\QQ
%Li-Shan Huang -- Prob
Suppose that  $(X,Y)$ is uniformly distributed on the triangle:
\[  \{(x,y): x \geq 0, x+|y|\leq 1 \}. \]

\begin{parts}
\PP
Find the distribution of $Y^2$.
\PP
Find the  conditional density of $Y$ given $X=x$, and  
$E(Y^2|X=x)$ for $0 \leq x \leq 1$.
\PP Using b), or otherwise, find  $E(1-X)^2$.
\end{parts}


\QQ
%Lin -- Applied Stat
Consider the multiple linear regression $Y=X\beta +\epsilon$
where $Y\ (n\times 1)$, $X\ (n\times p')$, $\beta\ (p'\times 1)$,and
$\epsilon\ (n\times 1)$ with $p'=p+1$.  Assume that the rank of 
$X$ is $p'$ $(\le n)$.  Define $H=X(X^TX)^{-1}X^T=(h_{ij})$ as the
HAT matrix.  Many diagnostic methods have been proposed to evaluate the quality
of fit and the quality of prediction in regression.

\begin{parts}
\PP
List \underline{five} important statistics which are useful for 
such diagnostic purposes.  Give simple expressions for these 
statistics and briefly describe their use.
\PP
It is said that ``a subregion in the regression space where
prediction is expected to be poor is a subregion in which the fit of the
model is expected to be relatively good."  True or false?
Explain.
\end{parts}

\QQ 
%Huffer -- Inference
Define $Y \sim {\cal P}(\lambda)$ to mean that $Y$ has the Poisson
distribution with mean $\lambda$.  Let $Y_1, Y_2, \ldots, Y_n$ be
independent random variables with $Y_i \sim {\cal P}(e^{\beta_0 +
\beta_1 x_i})$ where $\beta_0, \beta_1$ are unknown parameters and $x_1,
x_2, \ldots, x_n$ are known fixed constants.

\begin{parts}

\PP Find a two-dimensional sufficient statistic for $(\beta_0,
  \beta_1)$.

  \PP The MLE for $(\beta_0, \beta_1)$ can be found as the solution of a
  system of two equations.  What are these equations?  (Try to write
  these equations as simply as possible, but do NOT attempt to solve
  them.)

\end{parts}


\QQ
%Sethuraman -- Probability 
\begin{parts}

\PP An insect lays $N$ eggs, where $N$ has a Poisson distribution with 
parameter $\lambda$. However, the environment is harsh, and the probability 
of an egg surviving is only $p$. Assume that the survival of eggs are 
independent. What is the distribution of $X$, the number of eggs that 
survive? 

\PP
As the environment gets worse, the insect increases the number of eggs it lays 
by increasing the Poisson parameter $\lambda$ to go to $\infty$, while the 
survival probability of an egg, $p$ decreases to $0$ such that $\lambda p 
\rightarrow \mu$. What is the limiting probability that there will be some 
surviving eggs, i.e., what is the limit of $P\{ X > 0\}$? 
\end{parts}

\QQ
%Hollander -- Inference
Let $X_1,\ldots, X_n$ be a random sample from $N(\mu,\sigma^2)$.
\begin{parts}
\PP 
  Find the maximum likelihood estimators $\hat \mu$, $\hat\sigma^2$
  of $\mu,\sigma^2$.
\PP
   Find $E(\hat\mu)$.
\PP
  Find $E(\hat\sigma^2)$.
\end{parts}

 
\QQ
%Leysieffer -- Dist Theory 
A coin, when tossed, has probability $p$ of coming up heads and 
probability $q=1-p$ of 
coming up tails.  The coin is tossed repeatedly.  Determine the
expected values of each of the following random variables:
\begin{parts}

\PP $U$ is the waiting time (number of tosses) until HT appears;

\PP $V$ is the waiting time until HH appears;

\PP $W$ is the waiting time until a string of $m$ heads
appears.

\end{parts}
Hint: In b) and c) condition on the time of first occurrence of tails
to obtain equations that can be solved for $E(V)$ and $E(W)$.
\end{document}


