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\handout{quiz3}{Quiz 3 Version 2}{May 8, 1996}

This is an OPEN BOOK Quiz: you may refer to books, handouts, or your own
course notes.  (Calculators are not allowed.)  You may use in your problem
solutions any of the mathematical results given in lectures, recitations,
or the course text.  
There are five problems of roughly equal weight to
complete within one hour and 20 minutes.  
Good luck!

\begin{problems}

\problem
A random variable $X$ has a {\em two-point distribution}
if $X$ takes on only two values, $a$ and $b$.  Let the respective
probabilities of $X=a$ and $X=b$ be $p$ and $q=1-p$.  In this problem
assume that $X$ has a two-point distribution with mean $0$ and standard
deviation~$\sigma$, and assume $a\le 0$, $b\ge 0$.

\bparts
\ppart Express $b$ as function of $a$, $p$, and $q$.
% \ppart Derive an upper bound on $P(X\ge L)$, if $\abs{X}\le 2L$.
% Give a two-point distribution that illustrates that your bound is tight.
\ppart Show that $Var(X) = -ab$. ({\em Hint:} use part (a).)
\ppart Derive an upper bound on $P(X\ge c\sigma)$, 
       for some real constant $c\ge 1$, and
       give a two-point distribution that illustrates that your bound is tight.
       ({\em Hint:} In this distribution $b=c\sigma$.)
\ppart State the upper bound you get on $P(X\ge c\sigma)$ by applying
       Chebyshev's Theorem (and ignoring the fact that $X$ has a two-point
       distribution), and compare with the bound obtained in the previous
       part.
\eparts


\problem 
Let $X$ a random variable such that $E[X] = \mu$ and $X\leq  2\mu$.
Give an upper bound on $\Pr[X< {\mu\over 2}]$.

\problem
A personnel director has two lists of applicants for jobs.  List 1 contains
the names of five women and two men whereas list 2 contains the names of two
women and six men.  A name is randomly selected from list 1 and added to
list 2.  A name is then randomly selected from the augmented list.  Given
that the second name selected is that of a man, what is the probability that
a woman's name was originally selected from list 1?

\problem
Show that $S(n,n-1) = {n \choose 2}$ where $S(m,k)$ denotes a Stirling
number of the second kind.

\problem
Define a ``5-card major'' in a 13 card bridge hand to be a hand with either
exactly 5 spades or exactly 5 hearts or both.  What's the probability of
getting a 5-card major?

\end{problems}
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