

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

\begin{slide}
\begin{itemize}
\item Clearly, a good Monte Carlo evaluation requires small $\sigma$
(i.e., small ``variance'' of $f$).

\item We discuss this in numerical methods notes.

\item Note that in any Monte Carlo calculation, one should calculate
the error as well as value of integral.

\item Estimation of $\sigma$ involves calculating
\[
\int\limits^b_a f^2 (x) p(x) dx = \left\langle f^2 (x) \right\rangle
\]

\item which is estimated using Central Limit Theorem as
\[
\frac{1}{n} \sum^n_{i=1} f^2 \left( x_i \right)
\]

\item so one should always accumulate ${\displaystyle \sum} f^2$ as
well as ${\displaystyle \sum} f$ when doing a Monte Carlo integral.
\end{itemize}
\end{slide}

