Definition (Joint probability mass function)

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 $$p_{x,y}(x,y):=P(X=x,Y=x) \ , \ x\in\mathscr{X}, \ y\in\mathscr{Y}$$

Definition (Joint probability density function)

$X$ and $Y$ are jointly continuous if there exists a non-negative function $$f:\mathbb{R}\times\mathbb{R}\to[0,\infty)$$ such that for any reasonable set $C\subset\mathbb{R}^2$ (measurable), we have $$P((X,Y)\in C)=\int\int_Cf(x,y)dxdy$$RVs $X$ and $Y$ are called jointly continuous and $f$ is their joint pdf.