\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

