Joint probability mass function ($p_{X,Y}(x,y)=\mathbb{P}(X=x,Y=y)$)

For discrete random variables $X,Y$, their joint probability mass function assigns a probability to each possible ordered pair. Its entries are nonnegative and sum to one. Marginalization gives the probability mass functions $p_X(x)={\sum }_y{p}_{X,Y}(x,y)$ and $p_Y(y)={\sum }_x{p}_{X,Y}(x,y)$. In a table with rows indexed by $x$ and columns by $y$, these are the row and column totals. The joint probability distribution records how the coordinates occur together, while the marginal distributions describe each separately.