Gaussian Error Probability Function
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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 ) {\displaystyle {\mathcal σ 4}(\mu ,\,\sigma ^ σ 3)} Parameters μ ∈ R — mean (location) σ2 error function integral > 0 — variance (squared scale) Support x ∈ R PDF 1 2 σ 2 error function calculator π e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac σ 0{\sqrt − 9\pi }}}\,e^{-{\frac {(x-\mu )^ − 8} error function table − 7}}}} CDF 1 2 [ 1 + erf ( x − μ σ 2 ) ] {\displaystyle {\frac − 2 − 1}\left[1+\operatorname − 0 \left({\frac 9{\sigma {\sqrt 8}}}\right)\right]} Quantile μ + σ 2 erf
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− 1 ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt 2}\operatorname 1 ^{-1}(2F-1)} Mean μ Median μ Mode μ Variance σ 2 {\displaystyle \sigma ^ − 8\,} Skewness 0 Ex. kurtosis 0 Entropy 1 2 ln ( 2 σ 2 π e ) {\displaystyle {\tfrac − 6 − 5}\ln(2\sigma ^ − 4\pi \,e\,)} MGF exp { μ t + 1 2 σ 2 t 2 } {\displaystyle \exp\{\mu t+{\frac − inverse error function 0 σ 9}\sigma ^ σ 8t^ σ 7\}} CF exp { i μ t − 1 2 σ 2 t 2 } {\displaystyle \exp\ σ 2 σ 1}\sigma ^ σ 0t^ μ 9\}} Fisher information ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) {\displaystyle {\begin μ 41/\sigma ^ μ 3&0\\0&1/(2\sigma ^ μ 2)\end μ 1}} 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.[1][2] The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed. The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-sh
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 Created, developed, complementary error function table and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Statistical Distributions>Continuous Distributions> History and Terminology>Wolfram
Error Function Python
Language Commands> Interactive Entries>Interactive Demonstrations> Normal Distribution A normal distribution in a variate with mean and variance
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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, physicists sometimes call it https://en.wikipedia.org/wiki/Normal_distribution 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 used by Laplace in 1783 to http://mathworld.wolfram.com/NormalDistribution.html 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 which case is normal with mean and variance (5) (6) with .
a continuous function which approximates the exact http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html binomial distribution of events. The Gaussian distribution shown is normalized so that the sum over all values of x gives a probability of 1. The nature of the gaussian gives a probability of 0.683 of being within one error function standard deviation of the mean. The mean value is a=np where n is the number of events and p the probability of any integer value of x (this expression carries over from the binomial distribution ). The error function calculator standard deviation expression used is also that of the binomial distribution. The Gaussian distribution is also commonly called the "normal distribution" and is often described as a "bell-shaped curve". If the probability of a single event is p = and there are n = events, then the value of the Gaussian distribution function at value x = is x 10^. For these conditions, the mean number of events is and the standard deviation is . Show Gaussian curve IndexDistribution functionsApplied statistics concepts HyperPhysics*****HyperMath *****Algebra Go Back Gaussian Distribution Function The full width of the gaussian curve at half the maximum is Show IndexApplied statistics concepts HyperPhysics*****HyperMath *****Algebra Go Back