# Standard normal distribution

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1.

mko-standard-normal-distribution · definition · v0.10.0



## 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

[Standard-normal symmetry with explicit named facts](https://ocme.evemisslab.com/en/derivations/derivation-mko-standard-normal-distribution-01/)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](https://ocme.evemisslab.com/en/mko/mko-continuous-real-distribution-function/) — Use the continuous CDF background.
- [Improper integral](https://ocme.evemisslab.com/en/mko/mko-improper-integral/) — Density normalization has an improper-integral obligation.
- [Real exponential function](https://ocme.evemisslab.com/en/mko/mko-real-exponential-function/) — Use the Gaussian exponential.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The normalization constant uses a square root.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Mean semantics provide background, not a finite-law proof of Gaussian moments.
- [Variance in finite probability models](https://ocme.evemisslab.com/en/mko/mko-variance/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/source13-20261018-standard-normal-distribution.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-standard-normal-distribution/index.md)