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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam 
(Part I of Written Exam)\\
Sunday,  January 6, 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 You have two samples and two measures $X$ and $Y$ on each
item in the samples.  You plot the two samples on the same
scatterplot, using + for one sample and O for the other.

Make three plots which depict the following situations:

\begin{parts}

\PP $X$ and $Y$ are strongly related, but the correlation is near zero.

\PP $X$ and $Y$ are not correlated within each sample, but, taken
as one combined sample, they are correlated.

\PP $X$ and $Y$ are correlated within each sample, but the
combined sample shows no correlation.

\end{parts}

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


\QQ Answer the following:

\begin{parts}
  
  \PP Why is it important to have all important predictor
  variables in a multiple regression model?  Compare the
  consequences of including worthless predictors in the model
  versus excluding important predictors from the model.
  
  \PP Discuss one aspect of how multicollinearity complicates
  multiple regression analyses.
  
  \PP Briefly discuss two major problems in the interpretation of
  the output from typical stepwise multiple regression programs.

\end{parts}

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


% xxx Note: Need an F-table for this problem.

\QQ An experiment was planned to compare three different
fertilizers (A, B, C) on water melon yields. The treatments were
randomly assigned according to a Latin square design conducted
over a large farm plot which was divided into rows and columns.
The watermelon yields (in tons per acre) were recorded below
after the growing season.

\begin{tabular}{cccc} \hline
 & \multicolumn{3}{c}{Column} \\ \cline{2-4}
Row & 1 & 2& 3  \\ \hline
1  & 9.5 (B) &  6.8 (C)  & 4.9 (A)   \\
2  & 7.9 (A) &  9.1 (B)  & 6.6 (C) \\
3  & 5.6 (C) &  7.6  (A) & 8.7 (B) \\ \hline
\end{tabular}


\begin{parts}
  
  \PP If you were the experimenter, how would you randomize the
  design? Write an appropriate additive model for the data and
  state your assumptions on the model.
  
  \PP Conduct an analysis of variance ($\alpha=0.05$) based on
  the data and draw your conclusions.

\end{parts}

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



\QQ We take four identical marbles. On the first we write the
symbols $A_{1}A_{2}A_{3}$, on the second we write $A_{1}$, on the
third we write $A_{2}$, and on the fourth we write $A_{3}$.  We
then put the four balls in an urn and select one at random.  Let
$B_{i},i=1,2,3$ denote the event that the symbol $A_{i}$ appears
on the selected marble.

Are the events $B_{1}$, $B_{2}$, $B_{3}$ mutually independent?
Are they pairwise independent?  Justify your answers.

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



\QQ Let $X_1, X_2, \ldots, X_n$ be independent and identically
distributed (i.i.d.) random variables having the common
distribution $F(x) = P(X \leq x)$.

\begin{parts}
  
  \PP Consider an arbitrary parameter $\theta = \theta(F)$.  What
  does it mean for an estimator of $\theta$ to be ``consistent''?
  
  \PP Suppose $F$ is the Poisson distribution with parameter
  $\lambda >0$.  Find a consistent estimator for $\lambda
  e^{-\lambda}$.
  
  \PP Verify that the estimator found in part (b) is indeed
  consistent.

\end{parts}

\QQ When a current $X$ (measured in amperes) flows through a
resistance $Y$ (measured in ohms), the power generated is given
by $W = X^2 Y$ (measured in watts).  Suppose that $X$ and $Y$ are
independent random variables having the probability densities
$f_X(x) = 6x(1-x)$ for $0 \leq x \leq 1$, and $f_Y(y) = 2y$ for
$0 \leq y \leq 1$.  Determine the probability density function of
$W$.

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


\QQ Each customer who enters the bookstore will buy a mathematics
book with probability $p$.  If the number of customers entering
the bookstore is Poisson distributed with mean $\lambda$, what is
the probability that the bookstore does not sell any mathematics
books?

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


\QQ Let $X_1, X_2, \ldots, X_n$ be a sample from a population
with the density function (pdf)
\begin{displaymath}
f(x\,|\,\theta) =
\frac{1}{2}\, e^{-|x-\theta|}\,, \;\;\;-\infty < x < \infty\,, \;\;
-\infty < \theta < \infty\,.
\end{displaymath}
Find the maximum likelihood estimator (MLE) of $\theta$.


\QQ Let $X_1, X_2, \ldots, X_n$ be a sample from a population
with the density function (pdf) given by
\begin{displaymath}
  f(x\, |\, \alpha, \beta, p) = \left\{
    \begin{array}{ll}
p \alpha e^{-\alpha x} & \mbox{for $x \geq 0$}\\
(1-p) \beta e^{\beta x} & \mbox{for $ x <0$}
    \end{array}
\right.
\end{displaymath}
where $0 < p < 1$ and $\alpha, \beta >0$.  Find a
sufficient statistic for $(\alpha, \beta, p)$.  (Your sufficient
statistic should be minimal, but you are {\bf not} required to
prove this.)



\end{document}

