

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

\begin{slide}
\begin{itemize}
\item Let's consider Monte-Carlo Integration---an important numerical
technique---from a formal point of view.

\item We want to evaluate
\[
I = \int\limits^b_a f(x) dx
\]
\item We can surely write
\[
I = (b-a) \int\limits^{+\infty}_{-\infty} f(x) p(x) dx
\]
\item where
\begin{eqnarray*}
p(x) & = & \frac{1}{(b-a)} \hspace{2em} a\leq x\leq b \\
& = & 0 \hspace{4em} \mbox{outside}\ [a,b].
\end{eqnarray*}
\item and
\[
{\displaystyle \int} p(x)\, dx = 1
\]
\end{itemize}
\end{slide}

