Analytical form of a joint PDF of two uniformly distributed random variables I was wondering if it is possible to write down (closed-form analytically) the joint density function of two uniformly distributed random variables ($X$ and $Y$, each on a closed interval $dX=x_2-x_1$ &amp; $dY=y_2-y_1$) as a function of the correlation coefficient. 

E.g. $f(X)=1/dX$ and $f(Y)=1/dY \Rightarrow$ support set of $f(X,Y)$ is between $(x_2,y_2)$ and $(x_1,y_1)$.

This picture on wikipedia shows the correlation on such a rectangle graphically: http://en.wikipedia.org/wiki/Correlation_and_dependence#mediaviewer/File:Correlation_examples2.svg

Is it possible to find a closed form analytical expression of $f(X,Y)$ as a function of the correlation coefficient? Furthermore, can we express $f(X+Y)$?
In general, you are talking about copulas. There are a lot of them, but they allows you to specify a joint distribution with uniform marginals.
$\frac{1}{(d(b-a)-c(b-a))}$ for two joint uniform random variables $X$ and $Y$ ranging from $[a,b]$ and $[c,d]$, respectively. You can check by integrating the formula to $1$.