

\begin{slide}
\subsection{\bf Joint Probability Distributions}
\begin{itemize}
\item We can now consider the important extension to more than one
variable.

\item Suppose we have two random variables $x$ and $y$ each taking
values in $\Re_1$.

\item Then we can conceive of a random variable $z$ which is the
ordered pair $(x,y)$.

\item Even if we know the probability distributions of $x$ and $y$
\begin{eqnarray*}
x & \rightarrow & p(x) dx \\
y & \rightarrow & q(y) dy
\end{eqnarray*}

\item The distribution of $z$ is {\em of course\/} not defined.  

\item Suppose that $z$ has density $r(x,y)$ so that probability that
$z$ lies in volume element $dx\,dy$ is $r(x,y) dx dy$.

\item A priori, all we know is that $r$ satisfies
\begin{eqnarray*}
\int\limits r(x,y)dy & = & p(x) \\
\int\limits r(x,y)dx & = & q(y)
\end{eqnarray*}

\item These are called marginal distributions (EDJRS p.~18).
\end{itemize}
\end{slide}

