\documentclass[12pt]{6042}
%\usepackage{eepic}
\usepackage{psfig}

\newcommand{\recn}[3]
{\setcounter{problemsetnum}{#1}
\handout{teamprobs#1}{Problems for Recitation #1 (#2/98)}{#3}
}


\begin{document}
 
\recn{12}{12/4}{December 4, 1998}

%\section{Team Problems}

\begin {problems}

\problem %1

The Petersburg Casino offers the following game: the gambler
bets a fixed wager, and then the dealer flips a fair coin
(dealers do not flip coins in US casinos, but they do in
Petersburg) until it comes up heads. The gambler receives
\$1 if the coin shows heads the first time, \$2
if it shows the first head at second toss, and in general
$\$2^{k-1}$ if the dealer tosses the coin $k$ times to get
the first head. The fixed wager is \$100,000.

\bparts

\ppart
What is the expected amount of money that the Petersburg Casino
is going to pay to a gambler that plays this game?
What about the variance?
Is it reasonable for the gambler to risk \$100,000
dollars in a game where there is a $50 \%$ chance of getting
only {\em one dollar} back (this is not really a mathematical
question: just think about that.)

\ppart
What is the probability that the gambler does not lose
money in this game, i.e. what is the probability that
he gets back from the Casino more than \$100,000?


\eparts

\problem %2

A Harvard student, lost in the underground corridors of MIT, 
finds himself in a room 
containing three doors.  Door 1 leads him to freedom (out of MIT) after one 
hour travel; door 2 returns him to the same room after four hours travel, and 
door 3 returns him to the room after seven hours travel.  
Each time he finds himself in the room, the Harvard student 
picks a door with equal probability.  
Let $T$ denote the time until he finds the exit. 

\bparts

\ppart Define a sequence of independent
identically distributed random variables $R_i$ and a stopping rule 
$Q$ such that

\begin{displaymath}
T = \sum_{i=1}^{Q}{R_i}
\end{displaymath}

\ppart Is $R_k$ independent of the event that $Q\geq k$?
Is $R_k$ independent of $Q$?

\ppart If it is applicable, use  Wald's Theorem to find $\expect{T}$.

\ppart Now suppose that an MIT student is also lost and finds herself in a 
similar situation to the Harvard student.  The MIT student never chooses the 
same door twice.  What is $\expect{T}$ for the MIT student?

\eparts


\iffalse

\problem

If a random variable has infinite expectation, then the variance
is undefined, however a random variable may have finite
expectation and infinite variance.

\bparts

\ppart 
Show an example of a random variable whose expectation is finite
but whose variance is infinite.

[It might be easier to use a random variable whose expectation is zero.]

\ppart
If you used a random variable which can assume negative values
in answering the previous question, try to find an example
of a {\em non-negative} random variable whose expectation is finite
and whose variance is infinite.

[Hint: notice that $\sum_{n=1}^\infty 1/2^{2n} =1/3$
and $\sum_{n=1}^{\infty} 1/2^n = 1$.]

\eparts


\problem

Let $X$ be a non-negative random variable with expectation
$n/2$. Markov's inequality implies that
\begin{equation}\label{eq:mark}
\pr{ X \geq \frac 23 n} \leq \frac 34 .
\end {equation}

\bparts

\ppart
Define a random variable $X$ with expectation $n/2$ such
that the bound~\eqnref{eq:mark} holds with equality.

\ppart If
$X_1,\ldots,X_n$ are a collection of {\em independent} unbiased Bernoulli
variables over the same sample space, then $X = \sum_i X_i$ has
expectation $n/2$ and is non-negative, therefore bound~\ref{eq:mark}
applies. Unfortunately it is exceedingly inaccurate.
However, under the general hypotheses for Markov's inequality
(where no independence is assumed) the bound~\eqnref{eq:mark}
is the best one can hope for.

Define a sample space and unbiasied Bernoulli variables $X_1,\ldots,X_n$
on it, such that for $X = \sum_{i=1}^n X_i$ the
  bound~\eqnref{eq:mark}
holds with equality (assume $n$ is a multiple of 3).

[Hint:  the variables will have to be dependent. 
The event $X_1= \cdots =X_n = 0$ has probability $1/4$
in the sample space.]

\eparts

\fi

\end {problems}

\end {document}
