---
tag: 0N57D1XY6T
kind: definition
name: Pairwise independence
msc: [60A05]
defines: 0ZB3M68NYQ
uses: [0P0MZ4X682]
sources:
  - url: https://en.wikipedia.org/wiki/Pairwise_independence
    locator: lead section
---

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(A_i)_{i \in I}$ be a family of events.
The family is pairwise independent if $A_i$ and $A_j$ are [[independent|0P0MZ4X682]] for all $i, j \in I$ with $i \ne j$, that is, $P(A_i \cap A_j) = P(A_i) \, P(A_j)$.

## Remarks

[[Mutual independence|0GCVEV6BJA]] implies pairwise independence, but the converse is false. The node [[pairwise independent events that are not mutually independent|0AR4B3K5QD]] gives an example.
The same definition applies to random variables: any two of them are independent.
Pairwise independence is enough for some results.
Examples are the formula for the variance of a sum and the weak law of large numbers with finite variance.

## Explanations

### L1 human

Pairwise independence checks the events two at a time.
It can happen that each pair of events is independent, but a group of three events is not.