#### Finding the pdf of $$Z = X + Y$$:

We are given the joint pdf. For instance in the case of the uniform distribution:

$f_{XY}(x,y)= \mathbb 1_{(0,a)}\times \mathbb 1_{(0,b)}$

and we want to find the pdf of $$f_Z(z)$$ where $$Z=X+Y$$.

$F_Z(z) = \Pr(Z \leq z) = \Pr(x+y \leq z)$