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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 10 -- Quiz Only!}
\date{The last day to take the quiz is Wednesday, May 19, 2004}
\maketitle

\noindent
Normal questions and advanced questions (A) are worth $5$ points; easy
questions (E) are worth $4$ points.  Complete either the easy, or the
advanced version, not both.

\begin{enumerate}

\item In a promotional quiz each player, independently, has the chance
of winning 1\,000 Euros with probability $p$.  It is obviously bad for
the organizer if more than one person wins, but it is also considered
bad for the promotion if nobody wins at all.
\begin{enumerate}
\item With 500 players participating, what is the optimal choice for
$p$, i.e.\ the one that maximizes the chance of exactly one player
winning?

\item For this value of $p$, what is the expected payout?
\end{enumerate}

\item Let $X$ be the number of emails you receive in any given hour,
and assume that $X$ is Poisson distributed with parameter $\lambda =
10\,\mathrm{h}^{-1}$.
\begin{enumerate}
\item What are the probabilities of receiving (i) no, (ii) exactly
six, (iii) at least eight email messages in this hour?
\item Use the approximation of the Poisson by a normal distribution to
re-answer the questions from part (a).
\end{enumerate}

\item Let $X_1$, $X_2$, \ldots\ be independent discrete random
variables taking values in $\{-\tfrac12, \tfrac12 \}$ with probability
function
\[
  p(-\tfrac12) = q\,, \qquad p(\tfrac12) = (1-q) \,,
\]
and $p(x)=0$ for all other values of $x$.
\begin{itemize}
\item[(E)] Find the expectation for the random variable
\[
  Z_N = \frac{X_1+\dots+X_N}N
\]
as $N \to \infty$.  

\item[(A)] Let $Y_n = 2^{-n} X_n$.  Find the expectation for the
random variable
\[
  Z_N = Y_1+\dots+Y_N 
\]
as $N \to \infty$.  
\end{itemize}
\end{enumerate}
\end{document}
