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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam\\
Monday,  August 16, 2004
\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

%%%%%%%%%%%%%%%%%%%% Applied Statistics %%%%%%%%%%%%%%%%%%%%%%


\QQ You work for a company that makes paper helicopters such as the
one you received with this examination. Your boss has told you to
design a better paper helicopter. (Better paper helicopters land
closer to the target when dropped from a height of six feet.)
You may modify the weight of the paper, the length and width of
all body parts, the amount of weight added to the helicopter (one
or two small paper clips), the color of the paper, the type of
paper, and any other parts of the helicopter that you think
useful to modify.  Describe how you would design and run an
experiment (or a series of experiments, if you prefer) to develop
a better paper helicopter.


\QQ The water flow from Wakulla Springs is being modeled.  The
response $y$ is discharge at the opening of the springs.
Hydrologists are attempting to relate this to three external
factors using multiple linear regression.  These three factors
are \vspace{.2in}

\noindent $x_1=$ amount of rainfall in previous 24 hours \\
\noindent $x_2=$ temperature of springs water \\
\noindent $x_3=$ level of local spray-field pumping (none, low, high) \\

The hydrologists assume that the only affect the pumping level
has on the discharge is to change the constant (intercept)
associated with the linear regression model, i.e., pumping is
additive.

\begin{parts}

\PP  Write out the multiple linear regression model for this analysis.  Clearly label your variables.

\PP The hydrologists wish to test their assumption on pumping.
Modify your answer to part (a) so that this test can be
accomplished.

\PP Using the model from part (b), give the expected mean
response at each of the three pumping levels, and state the null
and alternative hypotheses needed to test the assumption on
pumping levels.

\PP Under the assumption of normal errors with mean zero and
constant variance, the correct test statistic for this test is
distributed as an $F$ random variable.  What are the degrees of
freedom associated with $F$?
  
\end{parts}


\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}




%%%%%%%%%%%%%%%%%%%% Probability %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\QQ Let $X$ and $Y$ be independent Gaussian random variables 
with parameters 
$(\mu_1, \sigma_1^2)$ and $(\mu_2, \sigma_2^2)$, respectively.

\begin{parts}

\PP Use moment 
generating functions to {\it derive} the distribution of $Y - X$. 
Quote all the theoretical results that you are using.

\PP If $\mu_1 = 50.4$, $\mu_2 = 47.2$, $\sigma_1^2 = 37.7$ and 
$\sigma^2_2= 36.06$, compute $P( Y \geq X)$.

\item[Hint:] You can use without proof that the moment generating
  function of a Gaussian random variable with mean $\mu$ and
  variance $\sigma^2$ is $M(t) = e^{t\mu}e^{t^{2}\sigma^{2}/2}$.
  
\end{parts}



\QQ Let $X_1$ and $X_2$ be two independent random variables with
density functions $f_1(x)$ and $f_2(x)$, respectively. 

\begin{parts}
  
  \PP Derive the probability density function of the random
  variable $Y= X_1 + X_2$. (If you know the result but cannot
  prove it, just state it for partial credit and go on to the
  next part.)
  
  \PP Using this result, find the density function of $Y$, when
  $X_1$ and $X_2$ are two exponential random variables with
  intensities $\lambda_1$ and $\lambda_2$, respectively (that is,
  the means are $1/\lambda_1$ and $1/\lambda_2$, respectively).
  
\end{parts}

\pagebreak


\QQ Suppose $X$ and $Y$ are nonnegative, independent, continuous
random variables such that
\begin{displaymath}
P(Y>z) = (P(X > z))^\beta
\end{displaymath}
for all $z \ge 0$ where $\beta$ is a positive constant.  Find $P(X \le
Y)$.



%%%%%%%%%%%%%%%%%%% Theoretical Statistics %%%%%%%%%%%%%%%%%%%%


\QQ Suppose you observe a single observation $X$ from the Poisson
distribution with the pmf:
\begin{displaymath}
f(x \mid \theta)=\frac{\theta^x e^{-\theta}}{x!}.   
\end{displaymath}
where the unknown parameter $\theta >0 $.

\begin{parts}

\PP Find the MLE of $e^{-\theta}$.

\PP Is the MLE an unbiased estimator of $e^{-\theta}$ ?

\PP Can you come up with an unbiased estimator of $e^{-\theta}$?
  
\end{parts}


\QQ Let $X_1, X_2, \ldots, X_n$ be independent, identically
distributed random
variables having an exponential distribution with mean $\beta$.

\begin{parts}
  
  \PP Find the maximum likelihood estimator (MLE) of $\beta$ and
  {\it derive} its asymptotic (large sample) distribution.
  
  \PP Find the method of moments estimator (MOM) of $\beta$.
 
  \PP Let $X$ be a new independent observation from the same
  exponential distribution that generated the data $X_1, X_2,
  \ldots, X_n$. Assume that for these $n$ data points we observed
  $\bar{X} = 3.4$. Use the estimate of $\beta$ from either part
  (a) or (b) of the problem to approximate $P(X \leq 1)$.

  
\end{parts}


\QQ Let $X_1, X_2, \ldots, X_n$ be iid Bernoulli($p$) where $n
\geq 2$.  What is the best unbiased estimate of $p^{\vecc{2}}$?
Justify your answer.



\end{document}

