Cumulative 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 x e − t 2 d t = 2 π ∫ 0
Cumulative Error Calculation
x e − t 2 d t . {\displaystyle {\begin − 5\operatorname − 4 (x)&={\frac cumulative error distribution − 3{\sqrt {\pi }}}\int _{-x}^ − 2e^{-t^ − 1}\,\mathrm − 0 t\\&={\frac 9{\sqrt {\pi }}}\int _ 8^ 7e^{-t^
Cumulative Error Formula
6}\,\mathrm 5 t.\end 4}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − erf function calculator t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 1\operatorname 0 (x)&=1-\operatorname Φ 9 (x)\\&={\frac Φ 8{\sqrt {\pi }}}\int _ Φ 7^{\infty }e^{-t^ Φ 6}\,\mathrm Φ 5 t\\&=e^{-x^ Φ 4}\operatorname Φ 3 (x),\end Φ 2}} 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 error function table ) {\displaystyle \operatorname 1 (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 Φ 9 (x|x\geq 0)={\frac Φ 8{\pi }}\int _ Φ 7^{\pi /2}\exp \left(-{\frac Φ 6}{\sin ^ Φ 5\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 − 9\operatorname − 8 (x)&=-i\operatorname − 7 (ix)\\&={\frac − 6{\sqrt {\pi }}}\int _ − 5^ − 4e^ − 3}\,\mathrm − 2 t\\&={\frac − 1{\sqrt {\pi }}}e^ − 0}D(x),\end − 9}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 7 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) =
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Error Function Matlab
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 The best answers are voted up and rise to https://en.wikipedia.org/wiki/Error_function 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 $$\Phi(x) = P(N < x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt$$ (where $N \sim N(0,1)$ is a standard normal) by the expression $\erf(x) http://math.stackexchange.com/questions/37889/why-is-the-error-function-defined-as-it-is = 2\Phi(x \sqrt{2})-1$. 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 $\sqrt{2}$ floating around. probability statistics special-functions normal-distribution share|cite|improve this question asked May 8 '11 at 20:19 Nate Eldredge 49k356129 I had assumed it was because
ei pi SubscribeSubscribedUnsubscribe228228 Loading... Loading... Working... Add to Want to watch this again later? Sign in to add this video to a playlist. Sign in Share More Report https://www.youtube.com/watch?v=CcFUQhorgdc Need to report the video? Sign in to report inappropriate content. Sign in Transcript Statistics 16,499 views 44 Like this video? Sign in to make your opinion count. Sign in 45 6 Don't like this video? Sign in to make your opinion count. Sign in 7 Loading... Loading... Transcript The interactive transcript could not be loaded. Loading... Loading... error function Rating is available when the video has been rented. This feature is not available right now. Please try again later. Published on Nov 8, 2013This is a special function related to the Gaussian. In this video I derive it. Category Education License Standard YouTube License Show more Show less Loading... Autoplay When autoplay is enabled, a suggested video error function table will automatically play next. Up next Integral of exp(-x^2) | MIT 18.02SC Multivariable Calculus, Fall 2010 - Duration: 9:34. MIT OpenCourseWare 202,270 views 9:34 Evaluating the Error Function - Duration: 6:36. lesnyk255 1,783 views 6:36 Fick's Law of Diffusion - Duration: 12:21. khanacademymedicine 134,133 views 12:21 erf(x) function - Duration: 9:59. Calculus Society -ROCKS!! 946 views 9:59 Diffusion - Coefficients and Non Steady State - Duration: 23:29. Engineering and Design Solutions 10,952 views 23:29 Lecture 24 Fick's Second Law FSL and Transient state Diffusion; Error Function Solutions to FSL - Duration: 45:42. tawkaw OpenCourseWare 502 views 45:42 Error Function and Complimentary Error Function - Duration: 5:01. StudyYaar.com 11,854 views 5:01 Error or Remainder of a Taylor Polynomial Approximation - Duration: 11:27. Khan Academy 235,861 views 11:27 Video 1690 - ERF Function - Duration: 5:46. Chau Tu 566 views 5:46 Integration of Natural Exponential Functions - Duration: 16:58. ProfRobBob 22,053 views 16:58 Hyperbolic Sine and Cosine Functions (Tanton Mathematics) - Duration: 13:45. DrJamesTanton 13,324 views 13:45 The Gaussian Distribution - Duration: 9:49. S