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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 $dX=x_2-x_1$ & $dY=y_2-y_1$) 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 $f(X+Y)$?

Teddy38
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Mar123
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  • The facts that X and Y are uniform on known intervals and the value of their correlation rho do not determine their joint density (except when rho=+1 or rho=-1). – Did Oct 16 '14 at 15:40
  • Ok, I understand that. Is it possible then to find a (one of the infinitely many) closed form expression of f(X,Y)? (or directly f(X+Y), the one I am looking for) – Mar123 Oct 16 '14 at 15:55
  • You could consider the independent case. – Did Oct 16 '14 at 16:59

3 Answers3

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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.

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By running the following Matlab code I more or less understood copulas and the inverse cumulative function trick they use. But from reading the Wikipedia page I still don't understand how to find a closed form analytical one (if possible). (Sorry for starting a new answer, I was not able to comment and my guest account did not work) xh=1000; xl=500; dx=xh-xl; yh=400; yl=200; dy=yh-yl;

n = 100; rho = 0.8; nu = 1; T = mvtrnd([1 rho; rho 1], nu, n); U = tcdf(T,nu); X = [gaminv(U(:,1),2,1) tinv(U(:,2),5)]; figure plot(xl+U(:,1)*dx,yl+U(:,2)*dy,'.');

user185050
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  • Look at archimedian copulas. They have explicit forms. Take the mixed derivative of the copula to get the joint pdf. –  Oct 17 '14 at 12:23
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$\frac{1}{(d(b-a)-c(b-a))}$ 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 $1$.

Alex M.
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Gary Li
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