f(x,y) = x + y, for 0<x<1 and 0<y<1
f(x,y) = 0, otherwise
Derive the joint distribution function of X and Y.
The Attempt at a Solution
Using the definition, I obtained part of the joint distribution F(x,y) = (1/2)(xy)(x+y) for 0<x<1, 0<y<1. Leaving out the working for this as it is pretty standard. I am not sure if my next step is correct and so is hoping you guys can help check my logic...
1) F(x,y) has to be defined for all (x,y) in R^2. So I have to consider all possible points. Correct?
2) When 0<x<1 and y > or = 1, F(x,y) = left limit of F(x,y) as y tend to 1 = F(x,1) = (1/2)(x)(x+1)? I am guessing that values of y above 1 do not affect the joint distribution F(x,y) so it takes the same value as its left limit at the "boundary" of possible y values?
3) By symmetry, when x > or = 1 and 0<y<1, F(x,y) = (1/2)(y)(y+1).
4) Finally, for x > or = 1 and y > or = 1, F(x,y) = 1. And F(x,y) = 0 for all other values.
Phew. I am most concern about step 2). The rest were included for completeness. Thanks!!!