Product of finite probability distributions ($p(x,y)=p_1(x)p_2(y)$)

Given two finite probability distributions, assign the product of their probability masses to each ordered pair. The support is the Cartesian product of the two positive-mass supports. Distributivity gives ${\sum }_{x,y}{p}_1(x)p_2(y)=({\sum }_x{p}_1(x))({\sum }_y{p}_2(y))=1$. This is the joint distribution for independent coordinate outcomes; it is a construction from the marginals, not a claim that arbitrary jointly observed coordinates are independent.