---
tag: 0XCE070V8B
kind: definition
name: Independence of random variables
msc: [60A05]
defines: 0GQ938GDE0
uses: [0DKRR22AEQ, 0GCVEV6BJA, 09T8ZHST1V]
sources:
  - url: https://en.wikipedia.org/wiki/Independence_(probability_theory)
    locator: More than two random variables
---

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $I$ be an index set. For each $i \in I$, let $X_i$ be a [[random variable|0DKRR22AEQ]] with values in $(E_i, \mathcal{E}_i)$.
The family $(X_i)_{i \in I}$ is independent if for each finite nonempty set $J \subseteq I$ and each choice of $B_j \in \mathcal{E}_j$ for $j \in J$,
$$P\left(\bigcap_{j \in J} \{X_j \in B_j\}\right) = \prod_{j \in J} P(X_j \in B_j).$$

## Remarks

Equivalently, the σ-algebras $\sigma(X_i) = \{X_i^{-1}(B) : B \in \mathcal{E}_i\}$ are [[independent|09T8ZHST1V]].
For two random variables this is the [[independence of two random variables|0E9502NT92]].
Mutual independence implies pairwise independence, but the converse is false, as for [[events|0AR4B3K5QD]].
For real random variables $X_1, \dots, X_n$, independence holds if and only if the joint cumulative distribution function is the product of the marginal ones.

## Explanations

### L1 human

Independent random variables come from separate parts of an experiment that do not influence each other.
For $n$ tosses of a coin, the results of the tosses are independent random variables.