        \problemdata       % Takes 5 *mandatory* arguments
        {variance-6}             % latex-friendly label for the prob.
        {variance}               % The topic of the problem content.
        {Velleman}               % Source (if known)
        {S98 PS11-7; F97 PS11-8} % Usage (list ones you are aware of).
        {Theory Pig, S02}        % Last revision info (author, date).
% FA 2001, tutorial 13
\begin{problem}
 \textbf{Generalized Law of Large Numbers.}

The Weak Law of Large Numbers in Notes 13 was given for a sequence $G_1,
G_2, \dots$ of mutually independent random variables with the same mean
and variance.  We can generalize the Law to sequences of mutually
independent random variables, possibly with different means and variances,
as long as their variances are bounded by some constant.  Namely,

\begin{theorem*}
Let $X_1, X_2, \dots$ be a sequence of mutually independent random
variables such that $\variance{X_i} \leq b$ for some $b\geq 0$ and all
$i\geq 1$.  Let
\begin{align*}
\bar{X}_n & \eqdef (X_1 + X_2 + \cdots + X_n)/n, &
\mu_n & \eqdef \expect{\bar{X}_n}.
\end{align*}
Then for every $\epsilon > 0$,
\[
\pr{\abs{\bar{X}_n - \mu_n} \geq \epsilon} \leq \frac{b}{n\epsilon^2}.
\]
\end{theorem*}

\begin{problemparts}
\problempart Prove this Theorem.

\hint Adapt the proof (attached) of the Weak Law of Large Numbers from
Notes 13.

\solution{Essentially identical to the proof attached, except that
$\variance{G_i}$ gets replaced by $b$, and the equality becomes $\leq$
where the $b$ is first used.}

\problempart Conclude
\begin{corollary*}[Weak Law of Large Numbers]

For every $\epsilon > 0$,
\[
\lim_{n \rightarrow \infty}
        \pr{\abs{\bar{X}_n - \mu_n}  \geq \epsilon} = 0.
\]
\end{corollary*}

\solution{For fixed $\epsilon$, the bound $b/(n\epsilon^2)
\rightarrow 0$ as $n \rightarrow \infty$.}

\end{problemparts}
\end{problem}

