Let $Z_1$ and $Z_2$ be independent random variables with known distributions $F(.;\theta_1)$ and $F(.;\theta_2)$ of the same convolution closed family. Then $Y = Z_1 + Z_2$ has distribution $F(.;\theta_1 + \theta_2)$.
How can we derive the joint distribution $F_{Z_i,Y}(.,?)$ and density $f_{Z_i,Y}(.;?)$, $i=1,2$?