Copula Theory : CDF from Marginals

Question

I have given $(X,Y)$ to be a two-dimensional random vector with Exp(1)-marginals and a Copula

$C(u,v) = uv + (1-u)(1-v)uv$

I need to determine the density of $(X,Y)$, if it exists.

I would assume that it is the product of the density of the components. However, in the question it is not stated that the components are independent, so I am having my doubts. Could anyone clarify this please?

Answer 1

Using the definition of a copula, the joint distribution function is given by $F(x,y)=C(F_1(x),F_2(y))$, where the Exp(1) marginals are $F_1(x)=1-exp(-x)$ and $F_2(y)=1-exp(-y)$. Hence

$$F(x,y)=(1-e^{-x})(1-e^{-y})(1+e^{-x-y}).$$

The joint density is obtained as $f(x,y)=\frac{\partial ^2 F(x,y)}{\partial x\partial y}$