DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Variance of a finite real random variable

Definition

Let $(Ω, w)$ be a finite probability space: $Ω$ is a finite set, each weight $w (ω)$ is nonnegative, and $\sum_{ω \in Ω} w (ω) = 1$ . Let $X : Ω \to R$ be a real random variable and let $E [X] = \sum_{ω \in Ω} X (ω) w (ω)$ . Its variance is

$Var (X) : = E [(X - E [X])^{2}] = \sum_{ω \in Ω} (X (ω) - E [X])^{2} w (ω) .$

All sums are finite, so the definition requires no additional integrability or moment hypothesis.

Depends on

Dependency tree · 2 levels

Sources

Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, §4.1, Definition 4.1