Standard normal distribution

Mathematical statement

Gaussian-integral normalization, E[Z]=0 and Var(Z)=1 are explicitly named theorem facts, not derivations manufactured from finite quadrature. Exact families handle symbolic density, evenness and phi(x)/phi(0)=exp(−x^2/2), plus Phi(0)=1/2 and Phi(−x)=1−Phi(x) from declared normalization and symmetry. No general nonzero elementary CDF value is computed. Finite numerical integration proves neither normalization nor moments; a continuous CDF need not have a density or be normal.

Representative formula (exact TeX)

F(t)=P(Z\le t)

Assumptions

• Symbolic families fix the standard normal law without additional mu/sigma parameters.
• Normalization and moments retain explicit named-fact provenance and are not finite-computation results of this slice.

Symbol scope

• P — underlying probability measure
• Z — standard normal random variable
• F — continuous cdf with gaussian density
• t — real cdf argument

Intuition, definition, and misconceptions

Density qualification and the named normalization theorem are separate.

Evenness gives symmetric CDF identities under normalization.

Finite quadrature does not certify normalization or moments.

Notation candidate: P is the underlying probability measure and P(Z≤t) evaluates the measurable event {ω:Z(ω)≤t}; F(t) is the CDF at a real argument. This does not feed a random variable directly to a measure or substitute the random F(Z). Existing named-fact responsibilities for Gaussian normalization and moments are unchanged.

A few numerical integration points do not prove the Gaussian integral or moments, and an arbitrary continuous CDF is not a normal law.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Continuous real distribution function — Use the continuous CDF background.
• Improper integral — Density normalization has an improper-integral obligation.
• Real exponential function — Use the Gaussian exponential.
• Natural-number powers and principal square roots — The normalization constant uses a square root.
• Expectation in finite models — Mean semantics provide background, not a finite-law proof of Gaussian moments.
• Variance in finite probability models — Variance semantics provide background.

Verification boundary

Gaussian integration, exponential differentiation and integration by parts remain named theorem responsibilities. The shared law/CDF representative display needs final role-alignment review; it is not an ordinary equality of a probability measure and a CDF value.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add the finite-variance IID central limit theorem.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.