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# Gaussian Law Of Error Distribution

## Contents

For other uses, see Bell curve (disambiguation). Normal distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation N ( μ , σ 2 gaussian distribution function ) {\displaystyle {\mathcal μ 5}(\mu ,\,\sigma ^ μ 4)} Parameters μ

## Normal Distribution Formula

∈ R — mean (location) σ2 > 0 — variance (squared scale) Support x ∈ R PDF 1

## Multivariate Gaussian Distribution

2 σ 2 π e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac μ 1{\sqrt μ 0\pi }}}\,e^{-{\frac {(x-\mu )^ ∝ 9} ∝ 8}}}} CDF 1

## Normal Distribution Examples

2 [ 1 + erf ⁡ ( x − μ σ 2 ) ] {\displaystyle {\frac ∝ 3 ∝ 2}\left[1+\operatorname ∝ 1 \left({\frac ∝ 0{\sigma {\sqrt σ 9}}}\right)\right]} Quantile μ + σ 2 erf − 1 ⁡ ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt σ 3}\operatorname σ 2 ^{-1}(2F-1)} Mean μ Median μ Mode μ Variance σ normal distribution pdf 2 {\displaystyle \sigma ^ − 9\,} Skewness 0 Ex. kurtosis 0 Entropy 1 2 ln ⁡ ( 2 σ 2 π e ) {\displaystyle {\tfrac − 7 − 6}\ln(2\sigma ^ − 5\pi \,e\,)} MGF exp ⁡ { μ t + 1 2 σ 2 t 2 } {\displaystyle \exp\{\mu t+{\frac − 1 − 0}\sigma ^ ⁡ 9t^ ⁡ 8\}} CF exp ⁡ { i μ t − 1 2 σ 2 t 2 } {\displaystyle \exp\ ⁡ 3 ⁡ 2}\sigma ^ ⁡ 1t^ ⁡ 0\}} Fisher information ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) {\displaystyle {\begin − 51/\sigma ^ − 4&0\\0&1/(2\sigma ^ − 3)\end − 2}} In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include f

Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 http://mathworld.wolfram.com/NormalDistribution.html Created, developed, and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Statistical Distributions>Continuous Distributions> History and Terminology>Wolfram Language Commands> Interactive Entries>Interactive Demonstrations> Normal Distribution A normal distribution in a variate with mean and variance is a statistic distribution with probability density function (1) on the domain . While statisticians and mathematicians uniformly use the term "normal distribution" for this distribution, normal distribution physicists sometimes call it a Gaussian distribution and, because of its curved flaring shape, social scientists refer to it as the "bell curve." Feller (1968) uses the symbol for in the above equation, but then switches to in Feller (1971). de Moivre developed the normal distribution as an approximation to the binomial distribution, and it was subsequently gaussian law of used by Laplace in 1783 to study measurement errors and by Gauss in 1809 in the analysis of astronomical data (Havil 2003, p.157). The normal distribution is implemented in the Wolfram Language as NormalDistribution[mu, sigma]. The so-called "standard normal distribution" is given by taking and in a general normal distribution. An arbitrary normal distribution can be converted to a standard normal distribution by changing variables to , so , yielding (2) The Fisher-Behrens problem is the determination of a test for the equality of means for two normal distributions with different variances. The normal distribution function gives the probability that a standard normal variate assumes a value in the interval , (3) (4) where erf is a function sometimes called the error function. Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically or otherwise approximated. The normal distribution is the limiting case of a discrete binomial distribution as the sample size becomes large, in whic

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