Definition

For 2 RVs

Let $X,Y$ be two discrete RVs defined on the same sample space $S$ of a random experiment and taking values in the sets $\mathscr{X},\mathscr{Y}$. Then the joint pmf of $X$ and $Y$ is $$p_{x,y}(x,y):=P(X=x,Y=x) \ , \ x\in\mathscr{X}, \ y\in\mathscr{Y}$$ ## Proposition (Properties) 1. $p(x,y)\ge 0 \ \forall x\in\mathscr{X},y\in\mathscr{Y}$ 2. $p(x,y)=0 \ \forall x\notin\mathscr{X},y\notin\mathscr{Y}$ 3. $\sum_{x\in\mathscr{X}}\sum_{y\in\mathscr{Y}}p(x,y)=1$ 4. For $A\subset\mathscr{X}\times\mathscr{Y}$, $P((X,Y)\in A)=\sum_{(x,y)\in A}p(x,y)$