The question states:
Show that the pmf of any two discrete random variables is given by the discrete convolution of the individual pmfs.
I'm confused because everything I've seen online assumes independence. I was also confused because shouldn't the question instead say "show that the sum of the pmf of any two discrete random variables is given by the discrete convolution of the individual pmfs."
Does the question make sense?
Assuming that $X$ and $Y$ are discrete random variables then my solution is:
$Z=X+Y$
$$P(Z=z)=\sum_{x=-\infty}^{\infty}P(X=x,X+Y=z)=\sum_{x=-\infty}^{\infty} P(X=x, Y=z-x)=\sum_{x=-\infty}^{\infty} P(X=x)P(Y=z-x)=\sum_{x=-\infty}^{\infty} P(x)P(z-x)= \sum_{x=-\infty}^{\infty} p_x p_{y}(z-x)$$