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

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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.

Definition references

The standard normal density has total mass one

The indefinite integral of a nonnegative measurable function is a measure

The law of a random element is a probability measure

The Dirac set function at a point

Depends on

Used by

BAOAB underdamped Langevin splitting Algorithm
Bias-corrected and accelerated bootstrap interval Algorithm
Bias-corrected bootstrap interval Algorithm
Predictor-corrector sampling for score SDEs Algorithm
Unadjusted Langevin algorithm (ULA) Algorithm
Brownian paths have infinite total variation Corollary
Deterministic Ito integrals are Gaussian Corollary
Disjoint ideal fair-bit streams yield independent standard normals by Box–Muller Corollary
Distribution of a one-sided Brownian hitting time Corollary
Heat-semigroup martingales Corollary
Law of the Brownian maximum Corollary
Matching minimax rate in a Gaussian orthogonal sequence model Corollary
One-dimensional Brownian motion hits every point almost surely Corollary
Proper full conditionals of a nondegenerate bivariate normal law Corollary
The Brownian square martingale Corollary
The exponential Brownian martingale Corollary
Vector Levy characterization Corollary
A boundary null has a nonstandard likelihood-ratio limit Counterexample
A curved normal family can make a sufficient statistic incomplete Counterexample
A non-Gaussian posterior can have different MAP and mean Counterexample
A nonadapted step integrand breaks the Ito isometry Counterexample
A point-mass Gaussian prior excluding the truth prevents contraction Counterexample
A spiky scale prior defeats an arbitrary degenerate-Gaussian concentration rate Counterexample
Explicit Euler moments diverge for a cubic dissipative SDE Counterexample
Feller negligibility cannot be removed from the converse Counterexample
Gaussian measurement noise need not remain Gaussian at later PnP-ISTA inputs Counterexample
Infinite-dimensional Gaussian random-walk reverse flow is singular Counterexample
Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
State-dependent diffusion needs a divergence correction Counterexample
Strong Markov fails at a nonstopping random time Counterexample
The IID bootstrap can fail at the kink of the absolute mean Counterexample
The ordinary chain rule fails for Brownian motion Counterexample
The raw natural Brownian filtration need not be right-continuous Counterexample
Blom rank-normal scores for pooled draws Definition
Canonical group-sequential alpha-spending design Definition
Centered Borel Gaussian random element on a separable Banach space Definition
Chi-square laws, including zero degrees Definition
Empirical-Bayes credible-ball procedure in the inverse Gaussian sequence model Definition
Finite Gaussian mixture with positive-definite component covariances Definition
Gaussian Huber contamination model for covariance estimation Definition

…and 216 more results.

Dependency tree · two levels

29 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)