

\subsection{\bf Example of using Monte Carlo}
\subsection{\bf for Summations}

\begin{slide}
\begin{itemize}
\item Then computational complexity of whole problem is reduced by a
factor of 1000!  The price, of course, is that error is increased by
variance (over $l$) of $f_l$.  

\item This variance is added in to variance of anything else in
integrand including variance of $f_l$ over $x_a, x_b, x_c$ which is
already present.

\item Thus, we can increase total number of points $n$ (where $x_a,
x_b, x_c, l$ are chosen separately $n$ times) by 1000 and reduce
total error by $\sqrt{1000}\sim 32$ if we generate $l$ randomly
rather than using full calculation of sum.  

\item This strategy is often a big win!  

\item See following discussion for a related comment.
\end{itemize}
\end{slide}

