Pairwise independence

In probability theory , a pairwise independent set of random variables is a set of random variables, any pair of which is independent ^[1] . Any set of randomly independent random variables is pairwise independent, but not all pairwise independent sets are independent in the aggregate. Pairwise independent random variables with finite dispersion are not correlated .

In practice, if this is not taken out of context, it is considered that independence means independence in the aggregate . Thus, a sentence of the form “ $X$ {\ displaystyle X} $X$ , $Y$ ${\ displaystyle Y}$ $Y$ , $Z$ ${\ displaystyle Z}$ $Z$ are independent random variables "means that $X$ ${\ displaystyle X}$ $X$ , $Y$ ${\ displaystyle Y}$ $Y$ , $Z$ ${\ displaystyle Z}$ $Z$ are independent in aggregate.

Content

1 Example
2 Summary
3 See also
4 References

Example

Independence in aggregate does not follow from pairwise independence, as shown in the following example, attributed to S. N. Bernshtein ^[2]

Let random variables $X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ denote two independent coin flips. Put 1 denotes the loss of an eagle, 0 - tails. Let be $Z$ ${\ displaystyle Z}$ - a random variable equal to 1 if, as a result of exactly one of the two coin flips, an eagle fell out, and 0 otherwise. Then the three $(X,\;Y,\;Z)$ ${\ displaystyle (X, \; Y, \; Z)}$ has the following probability distribution :

$(X,Y,Z)=$ ${\ displaystyle (X, Y, Z) =}$	$\left\{{\begin{matrix}\ \\\ \\\ \\\ \end{matrix}}\right.$ ${\ displaystyle \ left \ {{\ begin {matrix} \ \\\ \\\ \\\ \ end {matrix}} \ right.}$	$(0,\;0,\;0)$ ${\ displaystyle (0, \; 0, \; 0)}$ with a probability of 1/4,
		$(0,\;1,\;1)$ ${\ displaystyle (0, \; 1, \; 1)}$ with a probability of 1/4,
		$(1,\;0,\;1)$ ${\ displaystyle (1, \; 0, \; 1)}$ with a probability of 1/4,
		$(1,\;1,\;0)$ ${\ displaystyle (1, \; 1, \; 0)}$ with a probability of 1/4.

Note that the distributions of each random variable individually are equal to: $f_{X}(0)=f_{Y}(0)=f_{Z}(0)=1/2$ ${\ displaystyle f_ {X} (0) = f_ {Y} (0) = f_ {Z} (0) = 1/2}$ and $f_{X}(1)=f_{Y}(1)=f_{Z}(1)=1/2$ ${\ displaystyle f_ {X} (1) = f_ {Y} (1) = f_ {Z} (1) = 1/2}$ . The distributions of any pairs of these quantities are also equal: $f_{X,\;Y}=f_{X,\;Z}=f_{Y,\;Z}$ ${\ displaystyle f_ {X, \; Y} = f_ {X, \; Z} = f_ {Y, \; Z}}$ where $f_{X,\;Y}(0,\;0)=f_{X,\;Y}(0,\;1)=f_{X,\;Y}(1,\;0)=f_{X,\;Y}(1,\;1)=1/4.$ ${\ displaystyle f_ {X, \; Y} (0, \; 0) = f_ {X, \; Y} (0, \; 1) = f_ {X, \; Y} (1, \; 0) = f_ {X, \; Y} (1, \; 1) = 1/4.}$

Since each of the pairwise joint distributions is equal to the product of the corresponding marginal distributions, the random variables are pairwise independent:

$X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ independent
$X$ ${\ displaystyle X}$ and $Z$ ${\ displaystyle Z}$ independent
$Y$ ${\ displaystyle Y}$ and $Z$ ${\ displaystyle Z}$ are independent.

In spite of this, $X$ ${\ displaystyle X}$ , $Y$ ${\ displaystyle Y}$ and $Z$ ${\ displaystyle Z}$ are not independent in aggregate because $f_{X,\;Y,\;Z}(x,\;y,\;z)\neq f_{X}(x)f_{Y}(y)f_{Z}(z)$ ${\ displaystyle f_ {X, \; Y, \; Z} (x, \; y, \; z) \ neq f_ {X} (x) f_ {Y} (y) f_ {Z} (z)}$ . For $(X,\;Y,\;Z)=(0,\;0,\;0)$ ${\ displaystyle (X, \; Y, \; Z) = (0, \; 0, \; 0)}$ the left side is 1/4, and the right is 1/8. Moreover, any of the three random variables $X$ ${\ displaystyle X}$ , $Y$ ${\ displaystyle Y}$ and $Z$ ${\ displaystyle Z}$ is uniquely determined by two others and equals their sum taken modulo 2 .

Summary

Generally for any $n\geqslant 2$ ${\ displaystyle n \ geqslant 2}$ can talk about $n$ ${\ displaystyle n}$ -ar independence. The idea is similar: a set of random variables is $n$ ${\ displaystyle n}$ -ar independent if any subset of its power $n$ ${\ displaystyle n}$ is independent in aggregate. $n$ ${\ displaystyle n}$ -ary independence was used in theoretical computer science to prove the theorem on the MAXEkSAT problem.

Links

↑ Gut, A. Probability: a Graduate Course. - Springer-Verlag, 2005. - ISBN 0-387-27332-8 . pg. 71-72.
↑ Hogg RV, McKean JW, Craig AT Introduction to Mathematical Statistics. - 6. - Upper Saddle River, NJ: Pearson Prentice Hall, 2005. - ISBN 0-13-008507-3 . Remark 2.6.1, p. 120.

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

[2] Hogg RV, McKean JW, Craig AT Introduction to Mathematical Statistics. - 6. - Upper Saddle River, NJ: Pearson Prentice Hall, 2005. - ISBN 0-13-008507-3 . Remark 2.6.1, p. 120.

Pairwise independence

Content

Example

Summary

See also

Links

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