# Mutual Information of Correlated Bivariate Uniform Distribution

We have correlated bivariate uniform distribution, where X and Y have a correlation coefficient $\rho$ and they uniformly distributed in the following rectangle. What is the mutual information of $X$ and $Y$?

Note that this shape is obtained by transforming the bivariate uniform distribution which $X_1$, $Y_1$ are uniformly distributed in the $[0,1]^2$. The transformation is as follows:

$U=[X_1,Y_1] = uniform-rand(2,n)$

$\begin{gathered} C = \left[ {\begin{array}{*{20}{c}} 1&\rho \\ \rho &1 \end{array}} \right] \\ V = eigenVector[C] = \left[ {\begin{array}{*{20}{c}} { - 1}&1 \\ 1&1 \end{array}} \right] \\ E = eigenValue[C] = \left[ {\begin{array}{*{20}{c}} {1 - \rho }&0 \\ 0&{1 + \rho } \end{array}} \right] \\ \end{gathered}$

$W = V*Sqrt(E)$

$[X,Y] = (W*U)'$

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What is stopping you in the computation of $I(X,Y)=\mathbb E(\log(f(X,Y)/(f_X(X)f_Y(Y))))$? – Did Oct 28 '12 at 7:48
@did: I tried to compute it, but I could not find a correct solution. I found MI=0. However it is wrong. – PhD student Oct 28 '12 at 23:21