DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Moments, variance, and covariance on a probability space

Definition

Let $X$ be a real random variable on a probability space.

For $p > 0$ , the $p$ th absolute moment of $X$ exists when $∣ X ∣^{p}$ is integrable, and is then $E [∣ X ∣^{p}] .$
If $X$ is integrable, its mean is $E [X]$ .
If $X$ is square-integrable, its variance is $Var (X) : = E [(X - E [X])^{2}] .$

If $X$ and $Y$ are square-integrable real random variables on the same probability space, their covariance is $Cov (X, Y) : = E [(X - E [X]) (Y - E [Y])] .$

Depends on

Expectation of a nonnegative or integrable random variable

Moments, variance, and covariance on a probability space

Definition

Depends on

Dependency tree · 2 levels

Sources