

\begin{slide}
\subsection{\bf Example of Joint Probability}
\subsection{\bf Distributions: Bayes Law}
\subsection{\bf for Densities---I}
\begin{itemize}
\item Let event $X$ be $x$ lies in [x, x+dx]

\item Let event $Y$ be $y$ lies in [y, y+dy]

\item Let $T(X/Y) = t(x|y)dx$ be probability of $X$ given $Y$.  

\item By definition
\[
T(X/Y) = \frac{Pr (XY)}{Pr (Y)} = \frac{r(x,y)}{q(y)}\ \frac{dx\,dy}{dy}
\]

\item or $t(x|y) = r(x,y)/q(y)$ a conditional distribution (EDJRS p. 18).

\item Similarly, let $S(Y/X) = s(y|x)dy$ be probability of $Y$ given $X$.  

\item We find
\[
s(y|x) = \frac{r(x,y)}{p(x)}
\]
\end{itemize}
\end{slide}

