Gauss Error Function Wiki
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In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x complementary error function standard deviations above the mean. If the underlying random variable is
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y, then the proper argument to the tail probability is derived as: x = y − μ
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σ {\displaystyle x={\frac {y-\mu }{\sigma }}} which expresses the number of standard deviations away from the mean. Other definitions of the Q-function, all of which are simple transformations
Inverse Error Function
of the normal cumulative distribution function, are also used occasionally.[3] Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics. Contents 1 Definition and basic properties 2 Values 3 Generalization error function matlab to high dimensions 4 References Definition and basic properties[edit] Formally, the Q-function is defined as Q ( x ) = 1 2 π ∫ x ∞ exp ( − u 2 2 ) d u . {\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.} Thus, Q ( x ) = 1 − Q ( − x ) = 1 − Φ ( x ) , {\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,} where Φ ( x ) {\displaystyle \Phi (x)} is the cumulative distribution function of the normal Gaussian distribution. The Q-function can be expressed in terms of the error function, or the complementary error function, as[2] Q ( x ) = 1 2 ( 2 π ∫ x / 2 ∞ exp ( − t 2 ) d t ) = 1 2 − 1 2 erf ( x 2 ) -or- = 1 2 erfc ( x 2 ) . {\displaystyle {\begin{aligned}Q(x)&={\frac {1}{2}}\left({\frac {2}{\sqrt {\pi }}}\int _{x/{\sqrt {2}}}^{\infty }\exp \left(-t^{2}\right)\,dt\r
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, erf(inf) developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex erf(1) Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc error function python is the complementary error function, commonly denoted , is an entire function defined by (1) (2) It is implemented in the Wolfram Language as Erfc[z]. Note that some authors https://en.wikipedia.org/wiki/Q-function (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . For , (3) where is the incomplete gamma function. The derivative is given by (4) and the indefinite integral by (5) It has the special values (6) (7) (8) It satisfies the identity (9) It has definite integrals (10) (11) (12) For , is bounded http://mathworld.wolfram.com/Erfc.html by (13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above. A generalization is obtained from the erfc differential equation (14) (Abramowitz and Stegun 1972, p.299; Zwillinger 1997, p.122). The general solution is then (15) where is the repeated erfc integral. For integer , (16) (17) (18) (19) (Abramowitz and Stegun 1972, p.299), where is a confluent hypergeometric function of the first kind and is a gamma function. The first few values, extended by the definition for and 0, are given by (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). "Repeated Integrals of the Error Function." §7.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp.299-300, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp.568-569, 1985. Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; and Vetterling, W.T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in
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 http://mathworld.wolfram.com/Erfi.html Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Erfi https://www.scribd.com/document/50267654/Error-function-Wikipedia-the-free-encyclopedia Min Max Min Max Re Im The "imaginary error function" is an entire function defined by (1) where is the erf function. It is implemented in error function the Wolfram Language as Erfi[z]. has derivative (2) and integral (3) It has series about given by (4) (where the terms are OEIS A084253), and series about infinity given by (5) (OEIS A001147 and A000079). SEE ALSO: Dawson's Integral, Erf, Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfi/ REFERENCES: Sloane, N.J.A. Sequences A000079/M1129, A001147/M3002, and A084253 in gauss error function "The On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|Alpha: Erfi CITE THIS AS: Weisstein, Eric W. "Erfi." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Erfi.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org» Join the initiative for modernizing math education. Online Integral Calculator» Solve integrals with Wolfram|Alpha. Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal» Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Wolfram Language» Knowledge-based programming for everyone. Contact the MathWorld Team © 1999-2016 Wolfram Research,
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