Probability Values For Normal Error 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 , erf function σ 2 ) {\displaystyle {\mathcal σ 4}(\mu ,\,\sigma ^ σ 3)} Parameters error function calculator μ ∈ R — mean (location) σ2 > 0 — variance (squared scale) Support x ∈ R complementary error function table PDF 1 2 σ 2 π e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac σ 0{\sqrt − 9\pi }}}\,e^{-{\frac {(x-\mu )^ − 8} − gaussian function 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 − 1 ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt 2}\operatorname 1 ^{-1}(2F-1)} Mean μ Median
Normal Distribution Table
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 − 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
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Complementary Error Function Calculator
Interactive Entries>Interactive Demonstrations> Normal Distribution Function A normalized form of the cumulative normal inverse error function distribution function giving the probability that a variate assumes a value in the range , (1) It is related to the error function matlab probability integral (2) by (3) Let so . Then (4) Here, erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range is https://en.wikipedia.org/wiki/Normal_distribution therefore given by (5) Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so must be either computed numerically or otherwise approximated. Note that a function different from is sometimes defined as "the" normal distribution function (6) (7) (8) (9) (Feller 1968; Beyer 1987, p.551), although this function is less widely encountered than the usual . The notation is due http://mathworld.wolfram.com/NormalDistributionFunction.html to Feller (1971). The value of for which falls within the interval with a given probability is a related quantity called the confidence interval. For small values , a good approximation to is obtained from the Maclaurin series for erf, (10) (OEIS A014481). For large values , a good approximation is obtained from the asymptotic series for erf, (11) (OEIS A001147). The value of for intermediate can be computed using the continued fraction identity (12) A simple approximation of which is good to two decimal places is given by (14) The plots below show the differences between and the two approximations. The value of giving is known as the probable error of a normally distributed variate. SEE ALSO: Berry-Esséen Theorem, Confidence Interval, Erf, Erfc, Fisher-Behrens Problem, Gaussian Integral, Hh Function, Normal Distribution, Probability Integral, Tetrachoric Function REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp.931-933, 1972. Bagby, R.J. "Calculating Normal Probabilities." Amer. Math. Monthly 102, 46-49, 1995. Beyer, W.H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Bryc, W. "A Uniform Approximation to the Right Normal Tail Integral." Math. Comput. 127, 365-374,
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