Gaussian Distribution Error Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x error function calculator x e − t 2 d t = 2 π ∫ 0 error function table x e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi inverse error function }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted error function matlab erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ
Error Function Excel
6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),
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,
Error Function Python
and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire complementary error function table Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" complementary error function calculator encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by (1) Note that some authors (e.g., Whittaker https://en.wikipedia.org/wiki/Error_function and Watson 1990, p.341) define without the leading factor of . Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For , (5) where is http://mathworld.wolfram.com/Erf.html the incomplete gamma function. Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). For , (11) (12) Using integration by parts gives (13) (14) (15) (16) so (17) and continuing the procedure gives the asymptotic series (18) (19) (20) (OEIS A001147 and A000079). Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as (25) (26) Its derivative is (27) where is a Hermite polynomial. The first derivative is (28) and the integral is (29) Min Max Re Im Erf can also be extended to the complex plane, as illustrated above. A simple integral involving erf that Wolfram Language cannot do is given by (30) (M.R.D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include (31) (M.R.D'Orsogna, pers. comm., Dec.15, 2005). Erf has the continued fraction (32) (33) (Wall 1948, p.357), fir
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 http://mathworld.wolfram.com/NormalDistributionFunction.html updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at http://www.alglib.net/specialfunctions/distributions/normal.php WolframResearch Probability and Statistics>Statistical Distributions>Continuous Distributions> Interactive Entries>Interactive Demonstrations> Normal Distribution Function A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range , (1) It is related to the probability integral error function (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 therefore given by (5) Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and error function calculator 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 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 kn
one of the most known continuous distributions. Strictly speaking, there is a set of normal distributions which differs in scale and shift. Hereinafter, by "normal distribution" we imply so called standard normal distribution - normal distribution having mean equal to 0 and standard deviation equal to 1. Probability density of such normal distribution is: Cumulative distribution function is expressed using the special function erf(x): Algorithms Erf and ErfC subroutines are used to calculate the values of the special function erf(x) and its own complement. Inverse erf function is calculated by using the InvErf subroutine. Normal distribution's cumulative distribution function is calculated using the NormalDistribution subroutine. Inverse cumulative distribution function is calculated by using the InvNormalDistribution subroutine. This article is intended for personal use only.Download ALGLIB C# C# source. Downloads page C++ C++ source. Downloads page C++, multiple precision arithmetic C++ source. MPFR/GMP is used. GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org. Downloads page FreePascal FreePascal version. Downloads page Delphi Delphi version. Downloads page VB.NETVB.NET version. Downloads page VBAVBA version. Downloads page PythonPython version (CPython and IronPython are supported). Downloads page ALGLIB - numerical analysis library, 1999-2016. ALGLIB is a registered trademark of the ALGLIB Project. Policies for this site: privacy policy, trademark policy.