Error Function Of Normal Distribution
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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, moment generating function for normal distribution developed, and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Statistical Distributions>Continuous Distributions> Interactive Entries>Interactive
Parameters Of A Normal Distribution
Demonstrations> Normal Distribution Function A normalized form of the cumulative normal distribution function giving the probability that
Error Function Values
a variate assumes a value in the range , (1) It is related to the probability integral (2) by (3) Let so . Then (4) Here, erf is a
Erf Function
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 so must be either computed numerically or otherwise approximated. Note that a function different from is sometimes defined as error function calculator "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 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
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, and nurturedbyEricWeisstein at WolframResearch Calculus inverse error function and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... error function table History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which erf(inf) is a normalized form of the Gaussian function). It is an entire function defined by (1) Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor http://mathworld.wolfram.com/NormalDistributionFunction.html 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 the incomplete gamma function. Erf can also be defined as a Maclaurin series (6) http://mathworld.wolfram.com/Erf.html (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), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp.8-9). Definite integrals involving includ
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn http://math.stackexchange.com/questions/37889/why-is-the-error-function-defined-as-it-is more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer http://www.johndcook.com/blog/2008/03/15/error-function-and-the-normal-distribution/ site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer error function The best answers are voted up and rise to the top Why is the error function defined as it is? up vote 35 down vote favorite 6 $\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of course, it is closely related to the normal cdf $$\Phi(x) = P(N < x) = \frac{1}{\sqrt{2\pi}} error function of \int_{-\infty}^x e^{-t^2/2}dt$$ (where $N \sim N(0,1)$ is a standard normal) by the expression $\erf(x) = 2\Phi(x \sqrt{2})-1$. My question is: Why is it natural or useful to define $\erf$ normalized in this way? I may be biased: as a probabilist, I think much more naturally in terms of $\Phi$. However, anytime I want to compute something, I find that my calculator or math library only provides $\erf$, and I have to go check a textbook or Wikipedia to remember where all the $1$s and $2$s go. Being charitable, I have to assume that $\erf$ was invented for some reason other than to cause me annoyance, so I would like to know what it is. If nothing else, it might help me remember the definition. Wikipedia says: The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics. So perhaps a practitioner of one of these mysterious "other branches of mathematics" would care to enlighten me. The most reasonable expression I've found is that $$P(|N| < x) = \erf(x/\sqrt{2}).$$ This at least gets rid of all but one of the apparently spurious constants, but still has a peculiar
on 15 March 2008 by John The error function erf(x) and the normal distribution Φ(x) are essentially the same function. The former is more common in math, the latter in statistics. I often have to convert between the two.It's a simple exercise to move between erf(x) and Φ(x), but it's tedious and error-prone, especially when you throw in variations on these two functions such as their complements and inverses. Some time ago I got sufficiently frustrated to write up the various relationships in a LaTeX file for future reference. I was using this file yesterday and thought I should post it as a PDF file in case it could save someone else time and errors.Categories : Math StatisticsTags : Math Probability and Statistics Special functionsBookmark the permalink Post navigationPrevious PostWhat is the cosine of a matrix?Next PostConceptual integrity 7 thoughts on “Error function and the normal distribution” Blaise F Egan 8 October 2008 at 02:21 Very helpful. Thanks! jyotsna 21 November 2008 at 12:04 That was very useful ! thanks for your post ! 🙂 Theodore 9 December 2008 at 08:04 Greetings,Thanks for the post. Shouldn't the last term in the third equation in your pdf file be erf(x) and not erfc(x) ?Regards John 9 December 2008 at 08:47 Yes, you are right. Thanks for pointing out the error, no pun intended. I've corrected the file. Rasmus Bååth 3 October 2012 at 07:21 Great that you are posting this. Found it through google by searching for "error function density normal" 🙂 Diego Alonso Cortez 28 March 2013 at 12:24 Thank you sir! Richard 2 October 2015 at 07:07 many thx Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment Notify me of followup comments via e-mailName * Email * Website Search for: Subscribe to my newsletter Latest Posts Optimal team size Efficiency of C# on Linux GOTO Copenhagen Mathematical modeling for medical devices Publishable CategoriesCategoriesSelect CategoryBusinessClinical trialsComputingCreativityGraphicsMachine learningMathMusicPowerShellPythonScienceSoftware developm