

\begin{slide}
\subsection{\bf Example of Joint Probability}
\subsection{\bf Distributions: Bayes Law}
\subsection{\bf for Densities---II}
\begin{itemize}
\item {\em or\/} eliminating $r(x,y)$ from equations for $t(x|y)$ and
$s(y|x)$, we find
\begin{equation}
t(x|y) q(y) = s(y|x)p(x)
\end{equation}
\item This is a density version of Bayes law.

\item Example of a function of a random variable: 

\begin{description}
\item{$\bullet$} Let $y=f(x)$.  Now $y$ is a known function of $x$,
and clearly $s(y|x) = \delta (y-f(x))$.

\item{$\bullet$} Then using previous equation for $s(y|x)$, we can in 
this case find
\end{description}

\[
r(x,y) = \delta \left\lbrack y-f(x)\right\rbrack p(x).
\]
\end{itemize}
\end{slide}

