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) $F_{X,Y}(x,y)$ satisfies

F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),

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

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.

^ Gut, A. (2005) Probability: a Graduate Course, Springer-Verlag. ISBN 0-387-27332-8. pp. 71–72.
^ Hogg, R. V., McKean, J. W., Craig, A. T. (2005). Introduction to Mathematical Statistics (6 ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-008507-3.{{cite book}}: CS1 maint: multiple names: authors list (link) Definition 2.5.1, page 109.

[1] Gut, A. (2005) Probability: a Graduate Course, Springer-Verlag. ISBN 0-387-27332-8. pp. 71–72.

[2] Hogg, R. V., McKean, J. W., Craig, A. T. (2005). Introduction to Mathematical Statistics (6 ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-008507-3.{{cite book}}: CS1 maint: multiple names: authors list (link) Definition 2.5.1, page 109.

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