# Pairwise independence

﻿
Pairwise independence

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not independent.

Example

Here is perhaps the simplest example. Suppose "X", "Y", and "Z" have the following joint probability distribution:

:

Then

* "X" and "Y" are independent, and
* "X" and "Z" are independent, and
* "Y" and "Z" are independent, but
* "X", "Y", and "Z" are "not" independent, since any of them is just the mod 2 sum of the other two, and so is completely determined by the other two. That is as far from independence as random variables can get. However, "X", "Y", and "Z" are pairwise independent, i.e. in each of the pairs ("X", "Y"), ("X", "Z"), and ("Y", "Z"), the two random variables are independent.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Independence (probability theory) — In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. For example: The event of getting a 6 the first time a die is rolled… …   Wikipedia

• Independence of irrelevant alternatives — (IIA) is an axiom of decision theory and various social sciences. The word is used in different meanings in different contexts. Although they all attempt to provide a rational account of individual behavior or aggregation of individual… …   Wikipedia

• Statistical independence — In probability theory, to say that two events are independent, intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. For example:* The event of getting a 6 the first time a die is rolled …   Wikipedia

• Potentially all pairwise rankings of all possible alternatives — (PAPRIKA) is a method for multi criteria decision making (MCDM) or conjoint analysis based on decision makers’ preferences as expressed using pairwise rankings of alternatives. The PAPRIKA method – implemented via a specific type of… …   Wikipedia

• Arrow's impossibility theorem — In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives (options), no voting system can convert the ranked preferences of… …   Wikipedia

• Glossary of probability and statistics — The following is a glossary of terms. It is not intended to be all inclusive. Concerned fields *Probability theory *Algebra of random variables (linear algebra) *Statistics *Measure theory *Estimation theory Glossary *Atomic event : another name… …   Wikipedia

• Central limit theorem — This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 1 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result… …   Wikipedia

• List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

• Borel-Cantelli lemma — In probability theory, the Borel Cantelli lemma is a theorem about sequences of events. In a slightly more general form, it is also a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli.Let ( E n ) be a sequence… …   Wikipedia

• List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia