Joint distribution function

The joint distribution function is $F(x,y)=P(X\le x,Y\le y)$; mixed differentiation gives a joint density when sufficiently smooth.

A copula is a joint distribution function on a unit cube whose one-dimensional marginal distributions are uniform distributions. Applying each continuous marginal distribution to its random variable produces uniform marginals by the probability integral transform; their copula records the remaining dependence. Independent uniform coordinates and one common uniform coordinate give different copulas with the same marginals.