Finan 31.16 (but really need help with all convolution)

[Gooolati]

Hello all,

I am currently working on studying exam and had some questions regarding using convolution for the continuous case of the sum of two independent random variables. I have no problem with the actual integration, but what is troubling me is finding the bounds.


31.16 Consider two independent random variables X and Y. Let fx(x) = 1 - (x/2) for 0<=x<=2 and 0 otherwise. Let fy(y) = 2-2y for 0<=y<=1 and 0 otherwise. Find the probability density function of X+Y.



-for a = x+y

(fx*fy)(a) (the convolution) is the integral from negative infinity to infinity of fx(a-y)fy(y)dy or fy(a-x)fxdx


For 0<=a<=1 I integrate using the above formula and get

2a-(3/2)a^2 + (1/6)a^3

I would have thought the bounds to be from 0 to 1 but they are actually from 0 to a. And for the other cases of a (where 1<=a<=2 and 2<=a<=3) I am having trouble finding out where the bounds are. I am having trouble with basically all of these problems (discrete case as well) on finding the bounds and I really can't move forward until I fully understand so some insight would greatly be appreciated! Thanks!

[NoMoreExams]

Reading this thread may help: http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?t=257082&highlight=convolution