Finding a joint CDF F

Question

Given the joint density of two random variables $X$ and $Y$,

$f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$

How do I find the joint CDF ?

I know it'll be:

$F_{XY}(x,y)=\int\int_R f_{XY}(x,y)=\int\int_R2e^{-(x+y)}dxdy$ for $0<x<y$

I am unsure what my regions would be but I am guessing it is from x to infinity and y to infinity.

Answer 1

As we have $0<x<y<\infty$, if we take $x$ as dependent on $y$, we get - $$\iint_R f_{XY}dx dy= \int_0^\infty\int_0^yf_{XY}dxdy$$ If we would take $y$ as dependent on $x$, we get - $$\iint_R f_{XY}dx dy= \int_0^\infty\int_x^\infty f_{XY}dydx$$