Joint and Marginal Densities.

I am having some problems grasping the concept of these joint and marginal densities. It would really help if someone could provide me with an answer for the following question:

Find the joint and marginal densities corresponding to the cdf

F(X, Y) = (1 - е^αx){1-e^βy), x > 0, y>0, α > 0, β>0


TS Contributor
I think the joint CDF should be

\( F(x, y) = (1 - e^{-\alpha x})(1 - e^{-\beta y}), x, y > 0, \alpha, \beta > 0 \)

If you are familiar with exponential distribution, you can directly see that this
is the joint CDF of two independent exponential distribution.

Of course you may check with the following way easily:
The joint pdf \( f(x, y) = \frac {\partial^2 F(x, y)} {\partial x \partial y}
= \frac {\partial^2 F(x, y)} {\partial y \partial x} \)

The marginal pdf
\( f(x) = \int_0^{+\infty} f(x, y)dy, f(y) = \int_0^{+\infty}f(x, y)dx \)