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A copula is a multivariate distribution with uniform marginal distributions. Copulas are mostly used to represent or to model the structure of dependence between random variables, separately from the marginal distributions.

A copula is a multivariate distribution with uniform marginal distributions. Copulas are mostly used to represent or to model the structure of dependence between random variables, separately from the marginal distributions.

Let $F(y_1,...y_n)$ be the multivariate CDF of a random vector $Y$. We say the function $C(u_1,...,u_n)$ is the copula for the joint distribution of $Y$, when its marginals, $F(Y_j)$ are uniformly distributed.

The copula determines the distribution of every function of $Y$ that is invariant to univariate monotone transformations of $Y$. This means that the joint distribution of the ranks $r_1,...,r_n$ of an i.i.d. sample $Y_1, ..., Y_n$ from $F$ is entirely determined by $C$ (if the margins are absolutely continuous).

This is based on Sklar's theorem which shows that all multivariate distributions contain a copula, and how joint distributions are formed by coupling together marginal distributions with a copula. If you take a continuous multivariate distribution and apply the Probability Integral Transform to each margin, the resulting multivariate distribution has uniform margins and will be a copula.

Copulas are widely used in many application areas including finance, insurance, actuarial science, biostatistics, hydrology and weather research.