

\subsection{\bf Monte-Carlo Integration and}
\subsection{\bf the Central Limit Theorem---II}

\begin{slide}
\begin{itemize}
\item Let $\overline{I} = \frac{1}{n} {\displaystyle \sum^n_{i=1}}
f\left( x_i \right)$ where $x_i$ are distributed according to $p(x)$.

\item Now apply central limit theorem to $z_i = f\left( x_i \right)$.

\item $z_i$ are independent identically distributed random variables with 

\begin{description}
\item{$\bullet$} mean $\mu = {\displaystyle \int\limits^b_a} f(x) p(x)
dx$ and
\item{$\bullet$} variance $\sigma^2 = {\displaystyle \int\limits^b_a}
f^2 (x) p(x) dx - \mu^2$.
\end{description}

\item Then according to the Central Limit Theorem 

\begin{description}
\item{$\bullet$} $\overline{I} \rightarrow \frac{I}{(b-a)} \pm
\frac{\sigma}{\sqrt{n}}$
\end{description}

\item or $I$ can be estimated as $(b-a)\overline{I}$ with error $(b-a)
\frac{\sigma}{\sqrt{n}}$.
\end{itemize}
\end{slide}

