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3.2.3 Generalized Transformation Method:

Same as before, but with n-tuples of random variates:
Let $\mbox{$\bf x$}=(x_{1}, \dots ,x_{n})$, $\mbox{$\bf x$} \epsilon D_{x}$, and $\mbox{$\bf y$}=\mbox{$\bf f$}(\mbox{$\bf x$})$ with $\mbox{$\bf y$} \epsilon D_{y}$. Then
$\displaystyle p(\mbox{$\bf y$})$ $\textstyle =$ $\displaystyle p(\mbox{$\bf x$})
\left\vert \frac{\partial \mbox{$\bf x$}}{\partial\mbox{$\bf y$}}\right\vert$  

( $\vert\partial \mbox{$\bf x$}/ \partial\mbox{$\bf y$}\vert$ ... Jacobi determinant of the transformation $\mbox{$\bf x$}=\mbox{$\bf f$}^{-1}(\mbox{$\bf y$})$.)
\begin{figure}\includegraphics[height=180pt]{figures/f3tr3.ps}
\end{figure}
The following procedure for the production of Gaussian random variates may be understood as an application of this. $\Longrightarrow$

Normal distribution: 3.1



\fbox{ \begin{minipage}{600pt}
{\bf Box-Muller technique:}
\begin{itemize}
\item...
...iplying
$x_{1}$\ and $x_{2}$\ by their respective $\sigma_{i}$.
\end{minipage}}

Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001