

\subsection{\bf Interpolation to Find}
\subsection{\bf General Distributions---II}

\begin{slide}
\begin{itemize}
\item Then we can form the cumulative distribution
\end{itemize}

\vspace*{10pt}

\begin{tabular}{lll}
$x_i$ & 0 & $= F(x_1)$ \\
$x_1 + \delta x$ & $p(x_1)$ & $= F(x_1 + \delta x)$ \\
$x_1 + 2\delta x$ & $p(x_1) + p(x_2)$ & $= F(x_1 + 2\delta x)$ \\
$\vdots$ \\
$x_{m+1}$ & ${\displaystyle\sum^m_{i=1}} p(x_i)$ & = 1 if normalized 
$= F(x_{m+1})$
\end{tabular}

\vspace*{10pt}

\begin{itemize}
\item Then generate a random number $r$ uniformly with 
\[
0\le r\le 1
\]

\item Then use inverse interpolation to find $x$
\[
F(x) = r
\]

\item Then $x$ is distributed according to $p(x)$
\end{itemize}
\end{slide}

