TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A finite family of events is mutually independent exactly when its indicators are mutually independent

Statement

A finite family of events is mutually independent if and only if its indicator random variables are mutually independent. The same equivalence holds with pairwise independence in place of mutual independence.

Facts & Assumptions

Given: A finite family of events $(A_{i})_{i \in I}$ .

[L1]

Mutual independence is preserved when events are replaced by complements (Mutual independence is inherited by subfamilies and by replacing events with complements).

[L2]

The event ${1_{A} = 1}$ is $A$ , and ${1_{A} = 0}$ is $A^{c}$ (The indicator random variable of an event).

[L3]

Random variables are mutually independent exactly when all finite joint attained-value probabilities factor (Pairwise and mutual independence of finite-valued random variables).

Proof

technique · direct

1.1

Suppose the events are mutually independent. Every joint assignment $1_{A_{j}} = b_{j}$ , with $b_{j} \in {0, 1}$ , is an intersection of events $A_{j}$ and complements $A_{j}^{c}$ , whose probability factors by [L1].

L1L2

1.2

Conversely, if the indicators are mutually independent, specialize their joint-value identity to $b_{j} = 1$ for every chosen index; [L2] gives the event-intersection product identity.

L2L3

2.1

Hence the indicators are mutually independent by [L3].

step 1.1L3

3.1

Step 1.1 together with step 2.1 proves the forward direction, and step 1.2 proves the reverse direction. Restricting the same arguments to two indices proves the pairwise equivalence.

step 1.1step 1.2step 2.1∎

Depends on

Used by

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Sources

H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.4.1 and 3.1.5 (standard reference, not scraped)
J. Matousek and J. Vondrak, The Probabilistic Method, Sections 1.1 and 3.1 (standard reference, not scraped)