

\subsection{\bf Generation of General}
\subsection{\bf Random Number Distributions}
\begin{slide}
\begin{itemize}

\item Example:  Exponential Distribution

\item Suppose $x$ has distribution $e^{-x}$ for $x=0$ to $\infty$.
\begin{eqnarray*}
e^{-x} dx & = & d (e^{-x}) \\
y & = & e^{-x} \ \mbox{is uniform}
\end{eqnarray*}

\item Therefore, $x=-\ln y$ where $y$ is a uniform random variable in
range 0 to 1.

\item More generally put,
\[
p(x) dx = d(F(x))
\]

\item Then $w(x)=F(x)$, the characteristic function is uniformly
distributed
\[
w = \int\limits^x_{-\infty} p(x) dx
\]

\item (could be different lower limit if as above $p(x)=0$ where $x\le
0$).

\item You must be able to invert integral to be able to do this.
\end{itemize}
\end{slide}

