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\begin{document}


\begin{center}
  \bf \large
  M.\ S.\ Comprehensive Exam\\
  Friday, August 24, 2007
\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 Suppose a researcher wants to design a $2^{7-2}$ experiment with
factors $A, B, C, D, E, F, G$.  Upon further consideration, it was
determined that the last two factors, $F$ and $G$, must be assigned to
interactions involving the first five factors.

\begin{parts}

\PP Suppose the factor assignment of $F=ABC$ and $G=BCD$ is made.
Ignoring three-factor interactions and higher, list all of the
two-factor interaction aliases.

\PP What is the resolution of the design in (a) and explain how it can
be determined.

\PP Instead, suppose the factor assignment of $F=ABCD$ and $G=ABDE$ was
made.  Ignoring three-factor interactions and higher, list all of the
two factor interaction aliases.

\PP In general, would either of the designs of (a) or (c) be preferred
over the other?  Explain why or why not.
\end{parts}

\QQ The number of mishaps each week on an assembly line was
collected.

\begin{center}
\begin{tabular}{|c|c|}\hline
Number of Mishaps & Observed Frequencies \\
\hline
0 & 14 \\
1 & 28 \\
2 & 32 \\
3 & 25 \\
4 &  9 \\
5 &  6 \\
6+ & 2 \\
\hline
\end{tabular}
\end{center}

\begin{parts}
\PP You need to test if the data follows a Poisson
distribution with mean $\lambda=2$.  Use level of significance
$\alpha=.05$.  Give the hypotheses, the value of the test statistic,
and the criteria for rejecting the null hypotheses.

\PP If the mean of the Poisson distribution is not known,
but estimated to be $\lambda=2$, what changes are made to the answer
to part (a)?
  
\end{parts}

\pagebreak

\QQ Let $y=\beta_0 +\beta_1 x +\epsilon$ be a simple linear
regression, and the observations are $x_i, y_i, i = 1, \ldots, n$.
Let $\bar y$ denote the mean of $\{y_i\}$ and $\hat y_i$ denote the
least squares estimate of $y_i$ (i.e., the fitted value) by the linear
model.  Prove that:

$$R^2 = r^2$$

\noindent where $R^2$, the coefficient of determination, and $r^2$,
the correlation coefficient, are defined as
$$ R^2 = \frac{\sum (\hat y_i - \bar y)^2}{\sum (y_i - \bar y)^2}$$
and
$$ r = \frac{\sum (x_i - \bar x)(y_i - \bar y)}
{\sqrt{\sum (x_i - \bar x)^2}\sqrt{\sum (y_i - \bar y)^2}}\,\,.$$



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



\QQ Let the failure time of a component be $T$, a positive, continuous
random variable.  Let $F$ be the distribution (cdf) for $T$ and
suppose it may be written as $F(t)=1-e^{-H(t)}, t>0$, for some
differentiable function $H$.

\begin{parts}
\PP Show the density $f$ of $T$ satisfies
$$
\frac{f(t)}{1-F(t)}=H^\prime(t)\,.
$$
The function $h(t)\equiv H'(t)$ is referred to as the failure rate, or
hazard function, for $T$.

\PP Show for positive $t$ and $u$,
$$
P(T>t+u|T>t)=e^{-\int_t^{t+u}h(y)dy}
$$

\PP Suppose the hazard function $h$ for some random
variable $T$ is $h(t)=\theta$ for all $t>0$.  Find and name the
distribution of $T$.
  
\end{parts}


\QQ A {\it parallel} system is one that functions as long as at least
one of its components functions. A parallel system has $q$ independent
components, the i-th component having a life length with an
exponential distribution with mean $\beta_i$\, $(i =1, \dots, q)$. The
life length of the system is the maximum of the individual life
lengths of its components. What is the probability density function of
the life length of the system?


\QQ Answer the following.  (The parts are not related.)
\begin{parts}

  \PP If $Y$ is uniformly distributed over $(0, 5)$, what is the
  probability that the roots of the equation
$4x^2 + 4xY + Y + 2$
are both real?


\PP If $U$ is uniformly distributed over $(0, 1)$, find the density
function of $Z = e^{U}$.
  
\end{parts}

\pagebreak

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

\QQ Let $X_1,\ldots,X_n$ be an i.i.d. sample from an exponential
distribution with probability density function
\begin{eqnarray*}
    f_{\lambda}(x)=
    \begin{cases}
    \lambda e^{-\lambda x} & \text{if $x\ge0$,} \\
    0 & \text{otherwise.}\\
    \end{cases}
  \end{eqnarray*}


\begin{parts}
\PP
We want to test $H_0:\lambda=1$ versus $H_A:\lambda=2$.  Show  that
the Neyman-Pearson test rejects the null hypothesis for small values
of $2\sum_{i=1}^n X_i$.

\PP Find the distribution of $2 X_i$  under the null hypothesis
$H_0:\lambda=1$.

\PP It can be shown that
$2\sum_{i=1}^n X_i$ has a  $\chi_{2n}^2$ distribution under
$H_0:\lambda=1$.

Using the table for the percentiles of the $\chi^2$-distribution,
obtain the most powerful test for testing $H_0:\lambda=1$ versus
$H_A:\lambda=2$ at level $\alpha=.05$. Compute the rejection region
of this test.


\PP We observed $n=20$ observations with sample mean $\bar x=1.22$.
Test (at level .05) $H_0:\lambda=1$ versus $H_A:\lambda=2$ .
\end{parts}




\QQ Let $X_1,\ldots,X_n$ be iid Poisson($\mu$).

\begin{parts}

\PP Compute $p=P\{ X_1\ge 1\}$.

\PP Compute the maximum likelihood estimator $\widehat p$ of $p$.

\PP Compute the asymptotic variance of $\widehat p$.

\end{parts}


\QQ Find a sufficient statistic for $\theta$ in each of the following
situations.  (Your statistics should also be minimal, but you are {\bf
  not} required to show this.)

\begin{parts}
  
\PP You observe $X_1, \ldots, X_n$ iid from the density
\begin{displaymath}
  f(x \,|\, \theta) = 
  \begin{cases}
e^{x-\theta} & \mbox{for $x \leq \theta$}\\
0  & \mbox{for $x > \theta$\,.}
  \end{cases}
\end{displaymath}

\PP You observe independent random variables $X_1, \ldots, X_n$ where $X_i$ has
the density
\begin{displaymath}
  f_{X_i}(x \,|\, \theta) = 
  \begin{cases}
    e^{x-i\theta} & \mbox{for $x \leq i\theta$}\\
    0 & \mbox{for $x > i\theta$\,.}
  \end{cases}
\end{displaymath}


\end{parts}

\end{document}

