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Consider two continuous real valued random variables $X$ and $Y$. Let $f(X,Y)$ be their joint probability distribution and $f_X (X),f_Y(Y)$ their marginals. Suppose that $X$ and $Y$ are dependent. Is there any relation linking $f(X,Y)$ with the product $f_X(X)f_Y(Y)$, e.g. $f(X,Y)\leq f_X(X)f_Y(Y)$?

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  • $\begingroup$ Of course $f_X(x)$ and $f_Y(y)$ (hence their product) can be expressed in $f_{X,Y}(x,y)$, but that's all. Based on the inequality that you mention as example it can even be proved that $X$ and $Y$ are independent after all. $\endgroup$
    – drhab
    May 18, 2015 at 12:29

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As far as I know, there is no simple relation such as the one you propose in the case of dependant variables. But the relation between the joint distribution and the marginal distributions is described by a function called copula, which is the joint cdf of a random vector with uniform marginals.

http://en.wikipedia.org/wiki/Copula_%28probability_theory%29

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