DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Standard normal and normal laws

Definition

Assume AC. Define $γ (E) = \int_{E} e^{- x^{2} / 2} / \sqrt{2 π} d x$ for Borel E in $R$ . By The standard normal density has total mass one and The indefinite integral of a nonnegative measurable function is a measure, gamma is a probability measure; denote it $N (0, 1)$ . For $m \in R$ and $σ \geq 0$ , define $N (m, σ^{2})$ as the law of $x \mapsto m + σ x$ on $(R, B, γ)$ . This affine map is continuous: for $σ$ >0 choose $δ$ = $ε$ / $σ$ , and for $σ$ =0 it is constant. Its inverse images of opens are open, so it is Borel measurable. The law of a random element is a probability measure makes its pushforward a probability. When $σ$ =0, the preimage of E is all of R if m belongs to E and empty otherwise, so $N (m, 0) = δ_{m}$ in The Dirac set function at a point.

Depends on

Used by

Bayes formula for a finite mixture with continuous observation Example
Characteristic function of a gaussian law Example
Conditional density of a bivariate normal law Example
Independent sums via characteristic functions Example
Weak convergence of gaussian laws by parameters Example

Dependency tree · two levels

28 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

Durrett, Probability: Theory and Examples, Example 1.6.11, p.34; local normalization from the earlier published Gaussian integral (standard reference, not scraped)