

\subsection{\bf 1.5\hspace{1em} Properties of Joint}
\subsection{\bf Distribution Functions}

\begin{slide}
\begin{itemize}
\item (i) {\em Independence:}

\item If the distribution of $z$ satisfies
\begin{equation}
r(x,y) = p(x) q(y)
\end{equation}

\item then we say that $x$ and $y$ are statistically independent.

\item (ii) {\em Function of Two Variables}

\item Suppose, rather than $z$---an ordered pair in $\Re_2$, we
consider $h(x,y)$ in $\Re_1$: a conventional function of $x$ and $y$.

\item Then $w=h(x,y)$ is also a random variable with distribution $d(w)$ 
where
\begin{equation}
d(w) = \int\!\!\int dx\, dy\ \delta \left\lbrack w - h(x,y) 
\right\rbrack r(x,y)
\end{equation}

\item This may perhaps be proved very prettily: it is obvious if you
just do one---say $x$-integral in Equation~(8), and use the same
argument as already given for functions of a random variable.
\end{itemize}
\end{slide}

