DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pairwise independence

Definition

Let $(A_{i})_{i \in I}$ be a family of events in a probability space.

The family is pairwise independent when every two distinct members are independent, that is, when

$P (A_{i} \cap A_{j}) = P (A_{i}) P (A_{j}) (i \neq j) .$

Pairwise independence is weaker than independence of the whole family: it asks for the product rule only for two-member subfamilies.

Depends on

Independent sigma-algebras and independent events

Used by

Pairwise-independent Borel-Cantelli frequency law Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 2.1 (standard reference, not scraped)
S. R. S. Varadhan, Probability Theory, Section 3.1 (standard reference, not scraped)