# CopulasΒΆ

In this part, we will define the concept of copula.

To define the joined probability density function of the random input
vector by composition, one needs:

the specification of the copula of interest with its parameters,

the specification of the marginal laws of interest of the input variables .

The joined cumulative density function is therefore defined by:

Copulas allow to represent the part of the joined cumulative density
function which is not described by the marginal laws. It enables to
represent the dependency structure of the input variables. A copula is
a special cumulative density function defined on
whose marginal distributions are uniform on . The choice
of the dependence structure is disconnected from the choice of the
marginal distributions.

A copula, restricted to is a
-dimensional cumulative density function with uniform
marginals.

,

,

For all -box , we have , where:

, the summation being made over the vertices of .

- if
*for an even number of*,*otherwise*.

API:

See the list of available copulas.

Examples:

References:

Nelsen,

*Introduction to Copulas*Embrechts P., Lindskog F., Mc Neil A.,

*Modelling dependence with copulas and application to Risk Management*, ETZH 2001.