

\subsection{\bf The Wrong Way to Perform}
\subsection{\bf Multiple Monte Carlo}

\begin{slide}
\begin{itemize}
\item (v) When one does multiple integrals $\int dx\,\int dy\, f(x,y)$
by Monte Carlo and chooses a set of $n$ points $\underline{x}_i =
\left( x_i, y_i\right)$ where for $\underline{\rm each}$ $i$, one
chooses separately $x_i, y_i$ randomly.  This gives $\underline{x}_i$
distributed randomly over the plane.

\item It would be $\underline{\rm very}$ wrong to view as in Newton
Cotes as an integral over $x$ of $F(x)=\int dy\, f(x,y)$ where we think
of doing $x$ and $y$ integrals by Monte Carlo.

\item This incorrect approach gives a set of points $x_i, y_i$ where
one first chooses $x_1$ randomly, then fixing this value of $x_1$,
chooses $n_y\sim\sqrt{n}$ random values of $y$; then change to a new
random $x$ and so on.

\item This scenario gives use to a set $\left( x_i, y_i\right)$ where
\begin{eqnarray*}
x_1 & = & x_2 \ldots = x_{n_y} \\
x_{{n_y}+1} & = & x_{2n_y}\ \mbox{etc.} 
\end{eqnarray*}
\item In this case error is proportional to $1/\sqrt{n_x}\sim
1/n^{1/4}$ and NOT $1/\sqrt{n}$ where all $x_i$ and all $y_i$
are chosen independently.
\end{itemize}
\end{slide}

