**Context: statistics, probability, dependence**

To make things simpler, we shall first start off with two random variables. Later we can generalise this to *n* random variables.

Suppose you have two random variables, *X* and *Y*, which has the distribution function, *F*(*x*) and *G*(*y*), and has a joint distribution function *H*(*x*, *y*). Now by Sklar's Theorem, there exists a function *C*(*a*, *b*) (which satisfies the property of being a joint distribution function for two separate) random variables taking values in [0, 1]) such that

*H*(*x*, *y*) = *C*(*F*(*x*), *G*(*y*))

and the function *C* is called a copula. Its use is to couple two marginal distributions together to form a joint distribution.

Of course, when you replace the random variables *X* and *Y* with a random vector **x**, you get a *n*-dimensional form of Sklar's Theorem.

There is a more formal form of the definition of what a copula is, but the way described here should be sufficient for most practitioners.

Copulas are especially useful in studying dependence between random variables, something in which statisticians are always interested.