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.

Example

Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein.[3]

Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails. Let the third random variable Z be equal to 1 if exactly one of those coin tosses resulted in "heads", and 0 otherwise. Then jointly the triple (X, Y, Z) has the following probability distribution:

(X,Y,Z)=\left\{\begin{matrix} (0,0,0) & \text{with probability}\ 1/4, \\ (0,1,1) & \text{with probability}\ 1/4, \\ (1,0,1) & \text{with probability}\ 1/4, \\ (1,1,0) & \text{with probability}\ 1/4. \end{matrix}\right.

Here the marginal probability distributions are identical: f_X(0)=f_Y(0)=f_Z(0)=1/2, and f_X(1)=f_Y(1)=f_Z(1)=1/2. The bivariate distributions also agree: f_{X,Y}=f_{X,Z}=f_{Y,Z}, where f_{X,Y}(0,0)=f_{X,Y}(0,1)=f_{X,Y}(1,0)=f_{X,Y}(1,1)=1/4.

Since each of the pairwise joint distributions equals the product of their respective marginal distributions, the variables are pairwise independent:

However, X, Y, and Z are not mutually independent, since f_{X,Y,Z}(x,y,z) \neq f_X(x)f_Y(y)f_Z(z). Note that any of \{X,Y,Z\} is completely determined by the other two (any of X, Y, Z is the sum (modulo 2) of the others). That is as far from independence as random variables can get.

Generalization

More generally, we can talk about k-wise independence, for any k ≥ 2. The idea is similar: a set of random variables is k-wise independent if every subset of size k of those variables is independent. k-wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem MAXEkSAT.

See also

References

  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. Definition 2.5.1, page 109.
  3. ↑ 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. Remark 2.6.1, p. 120.