# Find joint pdf of uniformly distributed random variables

I have three random variables where $$a$$ and $$b$$ are uniformly distributed between $$[-1/2, 1/2]$$ and $$c = a + b$$. Also, $$b$$ is independent of $$a$$. How can I find the joint pdf of $$a$$ and $$c$$?

So far I've been looking at this formula:

Formula for calculating joint pdf

And I know that $$f_a = r_1(x)$$, the rectangular function with width 1. How can I calculate $$f_{c|a}$$?

• Have a look here This is a similar question with an answer. – callculus Feb 10 at 13:53
• @callculus Here the user is asking for the joint, so perhaps this one instead? Note that this question "intended" to ask for the marginal, while both answers there contain the joint. – Lee David Chung Lin Feb 10 at 14:11
• @LeeDavidChungLin Sure. But if I´m right the link I´ve posted the marginal is $f_X(z-y)=f_{X|Z}(z-y)$. Maybe I´m wrong. – callculus Feb 10 at 16:20