Convolution Method

A method to find PDF of PMF of Sum of Random Variables $Z=X+Y$ directly.

PDF

The PDF $f_Z(z)$ is given by: $f_Z(z)={\int }_{-\infty }^{\infty }{f}_{X,Y}(x,z-x)dx={\int }_{-\infty }^{\infty }{f}_{X,Y}(z-y,y)dy$

PMF

The PMF $p_Z(z)$ is given by: $p_Z(z)={\sum }_x{p}_{X,Y}(x,z-x)={\sum }_y{p}_{X,Y}(z-y,y)$