
\subsection{\bf Gaussian Limit of Binomial Distribution}

\begin{slide}
\begin{itemize}
\item Matthews and Walker confused me by discussing {\em two\/} limits
of binomial distribution.

\item We have already discussed one of these---the Poisson
limit---which is really a distinctive limit.  However, we can also see
a Gaussian limit which is just a special case of central limit
theorem.

\item Although this limit has a particular interpretation because of
the additivity property we just discussed.

\item Note also that central limit theorem was discovered in general in
1812 while de~Moivre discovered its binomial special case in 1733.

\item Let $x_i$ be independent random variables---each of which has the
same $b(k,1,p)$ distribution,

\item i.e., for each $i: x_i=0$ is failure and $x_i=1$ is success with
relative probabilities $1-p$ and $p$ respectively.

\item Then $y={\displaystyle \sum^N_{i=1}} x_i$ is in fact, $k$, the
number of successes in $n$ Bernoulli trials.

\end{itemize}
\end{slide}

