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

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