

\subsection{\bf Central Limit Theorem Statement}
\subsection{\bf Laplace}

\begin{slide}
\begin{itemize}
\item Let $y = \frac{1}{n} {\displaystyle \sum^n_{i=1}} x_i$:\ \ $x_i$
independent

\begin{description}
\item{$\bullet$ (a)} If $x_i$ are Gaussian, then so is $y$:\ \ in fact,
if $y$ Gaussian exactly, then so must all the $x_i$ be Gaussian.

\item{$\bullet$ (b)} De Moivre in 1733 showed that if $x_i$ binomially
distributed, then $y$ was always Gaussian for large $x$ (even though
Gauss didn't exist in 1733).

\item{$\bullet$ (c)} Laplace, in 1812, showed that de Moivre's result
was generally true for all distributions of $x_i$.
\end{description}

\item Central Limit Theorem of Laplace---Cramer p. 215

If $x_1 \ldots x_n$ are independent random variables with the same 
distribution, then $y$ is asymptotically Gaussian with mean $\mu$ and
standard deviation $\sigma/\sqrt{n}$ (where $x_i$ has mean $\mu$ and standard 
deviation $\sigma$, and both $\mu$ and $\sigma$ exist, i.e., are finite).
\end{itemize}
\end{slide}

