Q: Suppose X and Y are two continuous random variables whose joint density function is

f(x,y) = 2exp[-x-y] ( i.e 2e^(-x-y) ) , 0<x<y<+infinity

Find the corresponding Joint Cumulative Distribution Function.

I am having trouble with choosing my limits in the intergration.

obviously for: x<0 or y<0 or both F(x,y)=0 .

And for 0<x<y , x<y<inf (i.e the part above the line y=x in the first quadrant) i used the limits:

u=0 to x and v=0 to y

and this gives me: F(x,y) = 2exp[-x-y] -4exp[-x] +2

is this right?

and it is at this last part where i am having trouble choosing my limits of intergration where: 0<x<y and y<x<inf (the region below the line y=x in the first quadrant).

If anyone could tell me if what i have done so far is correct or not! it will be very helpful.

thanks