Pairwise independence

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.^[1] Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated.

A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) ${\displaystyle F_{X,Y}(x,y)}$ satisfies

${\displaystyle F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),}$

or equivalently, their joint density ${\displaystyle f_{X,Y}(x,y)}$ satisfies

${\displaystyle f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y).}$

That is, the joint distribution is equal to the product of the marginal distributions.^[2]

Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that independence means mutual independence. A statement such as " X, Y, Z are independent random variables" means that X, Y, Z are mutually independent.