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 \(Z=X+Y\).

We start with the distribution (cdf) function:

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

to later obtain the pdf through the derivative.

This is what would go on integrating under the bivariate uniform: