joint density problem

[shespuzzling]

I am having difficulty understanding a homework problem. I've attached what was given as the joint density of x & y (1 in the shaded area, 0 everywhere else).

We're asked to find F(z) and f(z) where Z=X+Y

I attempted to solve this by computing two double integrals, one for 0<z<1 and one for -1<z<0. For the first, I made y go from 0 to 1 and x go from 0 to z-y, and for the second, y went from -1 to 0, and x went from z-y to 0. The answer for each integral, respectively, was z-(1/2) and -z-(1/2).

I can't figure out how this would make sense, because adding the two gives you a negative number. Any help/guidance would be greatly appreciated.

Thanks.

[BGM]

Note when ,



Similarly when ,



Thus, when ,



when ,

[shespuzzling]

Thanks, I follow you, but I don't think I would have figured out to solve it that way on my own. Why can't you solve this by figuring out F(z) first (it makes more sense to me that way)? When you have a boundary for f(x,y) a triangle, like this problem, it seems to make more sense to me to find F(z) by integrating f(x,y) over the area with respect to z (i.e. for the positive triangle: the integral as y goes from 0 to 1, and the integral as x goes from 0 to z-y of f(x,y)). This gives you z-1/2, and then f(z) would be 1. I know that your answer is correct, but I can't figure out why my way isn't working....

Thanks again.