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\pset{10}{April 28, 1998}{May 5, 1998}

{\bf Reading assignment: 6.042 Fall 1997 Lecture Notes 21 and 22}

\begin{problems}
\problem{\bf [8 points]}

Show that if two positive random variables $R$ and $T$ are
independent, then $2R$ and $e^T$ are also independent.
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\problem{\bf [12 points]}

\bparts

\ppart{\bf [6 points]}
Let $R_1$ and $R_2$ be two independent random variables.
$R_1$ takes the values $0,1$ with equal probability.
$R_2$ has the following distribution.

$\Pr(R_2=1) = 1/2$\\
$\Pr(R_2=2) = 1/4$\\
$\Pr(R_2=4) = 1/4$

Compute the distribution of $2 ^{R_1} R_2$.

\ppart{\bf [6 points]}
Now suppose that $R_1$ and $R_2$ are no longer independent.
In particular, $R_1 = 0$ if $R_2$ is 1, and $R_1 =1$ when $R_2$ is 2 or 4.
$R_2$ has the same distribution as in part (a).

Compute the distribution of $2^{R_1} R_2$.

\eparts

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\problem{\bf [24 points]}

Consider the process of tossing a biased coin with
$\mbox{Prob(Heads)}=p$. Assume that all tosses are mutually independent.

\bparts
\ppart{\bf [6 points]}
What is the probability that exactly $k$ of the first 100 tosses will
turn up heads? 

\ppart{\bf [6 points]}
Consider the first 100 tosses. For what values of $k$ is the
probability of getting exactly $k$ heads greater than the probability
of getting exactly $k-1$ heads? For what values of $k$ is it less?
What value of $k$ is the most likely number of heads?

\ppart{\bf [6 points]}
Define $W_k$ to be the waiting time until the
$k$th head, that is, the number of tosses necessary to reach a total
of $k$ heads. For example, if the sequence of flips is {\em
HTTHHT...}, then $W_1=1, W_2=4,$ and $W_3=5$.

Determine $E(W_k)$ in terms of $k$ and $p$.  (Hint: determine $E(W_1)$
and use linearity of expectation.)

\ppart{\bf [6 points]}
Determine $\mbox{Prob}(W_k=n)$ in terms of $k$, $p$, and $n$.

\eparts

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\problem{\bf [15 points]}
Chuck-A-Luck is played in many casinos. One version of it is as
follows.  There is a cage with three dice in it.  The player pays
$\$1$ to guess a number between 1 and 6. The three dice are then
rolled. If one die shows the player's number, he wins $\$2$ (i.e., he
wins his old dollar back plus a dollar profit). If two dice show his
number, he wins $\$3$. If three dice show his number, he wins
$\$4$. If no die shows the number, he wins nothing.

If Bobo spends his last dollar on a game of Chuck-A-Luck, what is his
total expected wealth after the game?

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\problem{\bf [14 points]}
Consider a system consisting of $2$ components, each of which can
fail at each step with probability $p$.  Failures at different steps
and/or components are mutually independent (in particular, this means
that failure need not be permanent).  The steps are numbered $1,2,\ldots$.

\bparts 

\ppart{\bf [7 points]} Let $X$ be the random variable whose value is
the step number of the first step at which there is a failure (in
either component).  Find $E[X]$.

\ppart{\bf [7 points]} Let $Y$ be the random variable whose value is
the step number of the first step at which both components fail
simultaneously.  Find $E[Y]$.

\eparts
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\problem{\bf [15 points]} Suppose that we toss two hundred mutually
independent coins, each with Pr(Heads$)=0.4$

\noindent
Use the formula given in lecture to estimate the probability that 60
or fewer coins will be heads.

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\problem {\bf [12 points]} An airport bus deposits 8 passengers at 6
stops.  Assume that each passenger is as likely to get off at any stop
as another and that they act independently.  The bus stops only if
someone wants to get off.  What is the probability that nobody gets
off at the third stop?  What is the expected number of stops it will
make?

\end{problems}

\end{document}

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\problem Let $S$ be the set of functions from $\{1,\ldots,n\}$ to
$\{1,\ldots,k\}$, and let $\Pr:S\rightarrow [0,1]$ be the uniform
probability distribution on $S$, so that $\Pr(f)=1/|S|$ for all $f\in
S$.

Define the random variable $R\colon S\to \nat$ as $R(f) =
|\mbox{range}(f)|$, that is the random variable $R$ expresses the size
of the range of a function selected uniformly at random from $S$.

Express the expected value of $R$ as a function of $n$ and $k$.


\problem{\bf [5 points]}

We want to determine the percentage $r$ of people who intend to watch
the final episode of {\em Seinfeld.} Our polling method is the same as
that discussed in lecture: we sample the opinion of $n$ people chosen
uniformly and at random, with replacement.

How many people should be polled so that $r$ is within 3 points of the
true percentage, with probability at least .96?







