For the Gaussian case $I(X,Y)=f( \varrho )$ where $\varrho $ is the correlation coefficient, and $f$ is a known increasing function. Is there any known joint distribution where the $f$ is not strictly increasing, so for big values of $\varrho $ the $I(X,Y)$ might be small. Is there any way to construct such a distribution by applying a transformation on a random variable and by using a known distribution?
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$\begingroup$ I can come up with a mixture of Gaussians example, but that wouldn't be very elegant... $\endgroup$ – Memming Apr 19 '16 at 15:29

$\begingroup$ I would like to see it. Even a family of pmfs that has this property ( for the discrete case ) might be interesting. $\endgroup$ – Cauchy Apr 19 '16 at 15:38

$\begingroup$ I mean, it's not difficult to have high MI while CC is 0. Last row of en.wikipedia.org/wiki/… should give you some ideas. Sorry, too busy today to give you a detailed example. $\endgroup$ – Memming Apr 19 '16 at 22:28