density of sum of two independent uniform random variables

Question

I have two uniformly distributed random variables, $x\sim U[-a,a]$ and $y\sim U[-b,b]$ with $a>b>0$. How can I find the density function of $z=x+y$, which kind of different cases should I consider?

Answer 1

You have that: $$ Pr[x + y = c] = \sum_u Pr[x = u | y = c - u] = \sum_u Pr[x = u] \cdot Pr[y = c - u] $$ Plug in the known densities, and you are set. Might need to use integrals for continuous variables and give a range for the values, the idea is the same. The fact that the densities are defined over ranges makes it a bit messy: consider cases $a < b$ and $a \ge b$, or just $a < b$ (symmetrical with $a > b$) and $a = b$.