# Standard-normal symmetry with explicit named facts [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-standard-normal-distribution/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-standard-normal-distribution-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope Separate Gaussian prerequisites from derived CDF symmetry. No finite-quadrature proof of the Gaussian integral or moments. ## Givens and symbols - work - G1 — phi(x)=(2pi)^(-1/2)exp(-x²/2), Z has this density, and Phi(t)=P(Z≤t). - G2 — Explicit named theorem facts: the Gaussian integral is sqrt(2pi), hence normalization; E[Z]=0 and E[Z²]=1. These are premises, not proved here. - φ — standard-normal density - Φ — standard-normal CDF - Z — real random variable ## Rules and sources Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged. - mko_definition · cited — 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. · mko-standard-normal-distribution - work · prerequisite — The Gaussian integral and moments are explicit named theorem prerequisites; also assume integral linearity, substitution, atomlessness of density laws and the second-moment variance identity. ## Main derivation S01 · impliesBefore \varphi(x)=\frac{e^{-x^2/2}}{\sqrt{2\pi}} After \varphi(x)>0\quad(x\in\mathbb R) Conditions: none Rule: work · Uses: G1, G2, work Leaf basis: work The exponential and constant denominator are positive, qualifying nonnegativity. S02 · impliesBefore \int_{\mathbb R}e^{-x^2/2}\,dx=\sqrt{2\pi}\quad\text{(assumed)} After \int_{\mathbb R}\varphi(x)\,dx=1 Conditions: none Rule: work · Uses: G1, G2, S01, work Leaf basis: work Multiply the explicitly assumed Gaussian integral by the normalization constant. S03 · impliesBefore (-x)^2=x^2 After \varphi(-x)=\varphi(x) Conditions: none Rule: work · Uses: G1, G2, S02, work Leaf basis: work Evenness of the square gives pointwise evenness of the density. S04 · impliesBefore \varphi(0)=1/\sqrt{2\pi} After \varphi(x)/\varphi(0)=e^{-x^2/2} Conditions: none Rule: work · Uses: G1, G2, S03, work Leaf basis: work Cancel the common positive normalization constant to obtain the exact ratio. S05 · impliesBefore \Phi(-t)=\int_{-\infty}^{-t}\varphi(u)\,du After \Phi(-t)=\int_t^\infty\varphi(v)\,dv=1-\Phi(t) Conditions: none Rule: work · Uses: G1, G2, S04, work Leaf basis: work Substitute u=-v, using evenness and normalization; a density law has no singleton mass. S06 · impliesBefore t=0,\quad\Phi(0)=1-\Phi(0) After \Phi(0)=1/2 Conditions: none Rule: work · Uses: G1, G2, S05, work Leaf basis: work At zero the symmetry identity gives twice the CDF equal to one. S07 · impliesBefore E[Z]=0,\quad E[Z^2]=1\quad\text{(assumed)} After \operatorname{Var}(Z)=E[Z^2]-E[Z]^2=1 Conditions: none Rule: work · Uses: G1, G2, S06, work Leaf basis: work Substitute the explicitly assumed moments into the variance identity. S08 · impliesBefore t=1 After \Phi(-1)=1-\Phi(1),\quad\varphi(1)/\varphi(0)=e^{-1/2} Conditions: none Rule: work · Uses: G1, G2, S07, work Leaf basis: work Give exact symbolic relations at a nonzero point without inventing an elementary CDF value. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, G2, S07, work — Phi(1) has not been evaluated rationally, and finite integration proves none of the premises. - S07 · premise_audit · G1, G2, S06, work — Do not relabel the moment premises as an integral proof completed here. - S06 · premise_audit · G1, G2, S05, work — This exact value follows from symmetry and normalization, not approximation. - S05 · premise_audit · G1, G2, S04, work — The endpoint discrepancy in the complement has zero mass. - S04 · premise_audit · G1, G2, S03, work — This is a density ratio, not a CDF ratio. - S03 · premise_audit · G1, G2, S02, work — The center is 0; no unknown mu has been introduced. - S02 · premise_audit · G1, G2, S01, work — No finite quadrature points are being used to infer the integral. - S01 · premise_audit · G1, G2, work — Nonnegativity alone does not prove normalization. ## Error controls and negative controls Premise labelingKeep normalization and moments labeled as named facts, not newly proved results. Event and thresholdPhi(t)=P({omega:Z(omega)≤t}); t is a scalar threshold. - A few quadrature values prove normalization. → reject — Finite values do not prove an integral over the whole line. - Phi(1)=phi(1). → reject — Positive phi gives Phi(1)>Phi(0)=1/2, whereas phi(1)2, so they differ. - Every continuous CDF is standard normal. → reject — A uniform distribution has a different continuous CDF. ## Result and limitations Under the explicit Gaussian prerequisites, Phi(-t)=1-Phi(t), Phi(0)=1/2 and Var(Z)=1. ## Verification and source binding 2 declared checks for this guide; only the listed expressions are covered. - source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256: ce6e79a0422b1524888abb53a308a69a70552c942a0768637ab742837af70ae2 - original source SHA-256: 9be2b5a4eeeb9649020eca8625ccd6a76eef2d3e32963e884d8eadc5a4490a58 - Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW [Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-standard-normal-distribution-01.json)