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Canonical static concept record

Independent random variables

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Summary

A family of random variables whose finite subfamilies have joint probabilities that factor into their marginal probabilities.

Record metadata

Canonical name: Independent random variables
Concept ID: independent_random_variables
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Probability
Domains: Probability, Measure
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

common probability space
indexed family $(X_i)_{i\in I}$ of random variables

Axioms / constraints

for every finite $J\subseteq I$ , $\mathbb P(\bigcap_{j\in J}\{X_j\in A_j\})=\prod_{j\in J}\mathbb P(X_j\in A_j)$ for measurable $A_j$

Canonically induces

product finite-dimensional distributions

Notes

Pairwise independence is weaker than mutual independence and is not substituted here.

Concept sources

Wikipedia — Independence (probability theory) — Independence (probability theory) · encyclopedic reference · source ID wp-independent-random-variables
nLab — iid random variables — iid random variables · mathematical reference · source ID nlab-iid-random-variables
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Stochastic process Independent random variables

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: random_variable_to_independent_family
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + mutual independence

Authored explanation

Require every finite subfamily of the indexed random variables to have joint probabilities that factor into their marginals.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Independence (probability theory) — Independence (probability theory) · encyclopedic reference · source ID wp-independent-random-variables
nLab — iid random variables — iid random variables · mathematical reference · source ID nlab-iid-random-variables
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.