So suppose you have two discrete random variables $X$ and $Y$ and the goal is to find the distribution of $Z=X+Y.$
Now, I saw one formula which states that $$P(Z=N)=\sum_{k=0}^{N}P(X=N-k,Y=k).$$ I am not sure how to read $P(X=N-k,Y=k)$ in English. Is it $P(X=N-k)$ and $P(Y=k)$ except it is not since in the next line I see that $$P(Z=N)=\sum_{k=0}^{N}P(X=N-k,Y=k) = \sum_{k=0}^{N}P(X=N\cap Y=k)P(Y=k).$$ So why use the comma notation? What does it really mean?