Sum of discrete independent random variables

Question

in order to sum two discrete and independent random variables $X$,$Y$ do we need the cartesian product of the domains of $X$ and $Y$ ? If yes, what is the procedure to do so ?

Thanks :)

Answer 1

In general, you need the density of $(X,Y)$ on the cartesian product of the domains of $X$ and $Y$. Then the procedure would go as follows (define $Z=X+Y$) $$ \mathbb{P}(Z= z) = \mathbb{P}(X+Y=z) = \sum_{k}\mathbb{P}(X= k,\ Y= z-k) $$ where the sum is taken at least over all $k$ in the intersection of the domain of $X$.
However, if you have independence, the last term simplifies to $$ \sum_{k} \mathbb{P}(X=k)\mathbb{P}(Y=z-k) $$ and then you don't need it.