Given the functions
$$F(x,y) = \begin{cases}1-e^{-x-y}\quad \text{if} \ x,y \geq 0,\\ 0 \qquad \text{otherwise}.\end{cases}
$$
$$F(x,y) = \begin{cases}1-e^{-x}-xe^{-y}\quad \text{if} \ 0 \leq x \leq y,\\ 1-e^{-x}-ye^{-y}\quad \text{if} \ 0\leq y \leq x,\\0 \qquad \text{otherwise}.\end{cases}
$$
Check whether these are joint distribution of some pair $(X,Y)$. If they are, find the distribution of $X$ and $Y$ separately.
How would I tackle this?
I can do this if it's single variables, I would just differentiate and then check if it satisfies the PMF properties but I'm unsure for two variables.
Also, is the last part just finding the marginal densities?
In general a join cumulative distribution $F(x_1,x_2\cdots)$ must be monotonically non-decreasing for each of its variables $x_1,x_2,\cdots$. And this is not the case for the second case, here's a plot of $F$ as a function of $y$ for $x=1$