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Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Directions: MAKE SURE TO COPY YOUR ANSWERS TO A SEPARATE SHEET FOR SENDING
ME AN ELECTRONIC COPY LATER.

{\bf 1.} (15) Suppose $X$ and $Y$ are independent and have Poisson
distributions, then it can be shown that $S = X+Y$ also has a Poisson
distribution.  Fill the blank {\it with a term from our course}:  We say that
the Poisson family is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ under
independent summation.

{\bf 2.} Consider the class enrollment example, p.153.

\begin{itemize}

\item [(a)] (15) Give R code to evaluate Equation (7.24).

\item [(b)] (15) Give R code to find the upper 10\% point for class size,
i.e.\ a number above which only 10\% of class exceed.

\end{itemize}

{\bf 3.} Consider the toy population example, Sec. 9.2.1. Suppose we
take a \underline{simple} random sample of size 2.  Imagine a notebook
description of this, with columns labeled $X_1$, $X_2$ and
$\overline{X}$, and infinitely many lines.

\begin{itemize}

\item [(a)] (15) What is the number of distinct values in the
$\overline{X}$ column?

\item [(b)] (10) What is the long-run proportion of rows in which there is
a 72 in the $X_1$ column and a 69 in the $X_2$ column?

\item [(c)] (15) What is the long-run proportion of the value 72 in the
$X_2$ column?

\end{itemize}

{\bf 4.} (15) A dart is thrown at the interval (0,1).  The position $D$
that it hits is a random variable, with density $f_D(t) = 2t$ for $0 < t
< 1$ and 0 elsewhere.  Find the expected value of the distance from the
dart to the point 0.5.

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\newpage

{\bf Solutions:}

{\bf 1.} closed

{\bf 2.a}

\begin{lstlisting}
(1 - pnorm(30,28.8,3.1)) / (1 - pnorm(25,28.8,3.1)) 
\end{lstlisting}

{\bf 2.b}

\begin{lstlisting}
qnorm(0.90,28.8,3.1)
\end{lstlisting}

{\bf 3.a} 3

{\bf 3.b} 1/6

{\bf 3.c} 1/3

{\bf 4.} 

$$
\int_{0}^{1} |t - 0.5| ~ 2t ~ dt =
\int_{0}^{0.5} (0.5-t) ~ 2t ~ dt +
\int_{0.5}^{1} (t - 0.5) ~ 2t ~ dt
$$

\end{document}