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:

X and Y are independent, and
X and Z are independent, and
Y and Z are independent.

However, X, Y, and Z are not mutually independent, since $f_{X,Y,Z}(x,y,z)\neq f_{X}(x)f_{Y}(y)f_{Z}(z),$ the left side equalling for example 1/4 for (x, y, z) = (0, 0, 0) while the right side equals 1/8 for (x, y, z) = (0, 0, 0). In fact, 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.

k-wise independence is used in the proof that k-independent hashing functions are secure unforgeable message authentication codes.

References

↑ 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. Definition 2.5.1, page 109.
↑ 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.

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[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.

Pairwise independence

Example

Generalization

See also

References