$F(x,y) = 1-e^{-x}-e^{-y}+e^{-x-y}, (x,y) >= 0 $ and $0$ otherwise.
I found the joint pdf by differentiating by x and y to get $f(x,y) = e^{-x-y}$
However, when I find the marginal distributions of x and y they are both just 1. Is this valid? Or did I do something wrong here?
Another way to solve the problem is just to observe that
$$f_{XY}(x,y)=e^{-x-y}\mathbb{1}_{(0;+\infty)}(x)\mathbb{1}_{(0;+\infty)}(y)$$
can be factorized in
$$f_{XY}(x,y)=e^{-x}\mathbb{1}_{(0;+\infty)}(x)\times e^{-y}\mathbb{1}_{(0;+\infty)}(y)=f_X(x)\times f_Y(y)$$
Thus the two marginals are iid Exp(1)