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Topic: Regularized incomplete beta function

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	incomplete beta function (Site not responding. Last check: 2007-11-05)
		(a,b) is a variation on the beta function.
		Sometimes the function is normalized to the regularized incomplete beta function I(x,a,b), defined as:
		The incomplete beta function and its inverse are used in statistics.
	www.2dcurves.com /gamma/gammabi.html (33 words)

	Beta distribution - Wikipedia, the free encyclopedia
		In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]:
		The beta function is a normalization constant to ensure that the integral of the pdf is unity:
		Beta distributions are used extensively in Bayesian statistics, since the beta distribution is the conjugate prior distribution to the binomial distribution.
	en.wikipedia.org /wiki/Beta_distribution (230 words)

	Incomplete beta function - Wikipedia, the free encyclopedia
		In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral.
		The situation is analogous to the incomplete gamma function being a generalization of the gamma function.
		The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:
	en.wikipedia.org /wiki/Incomplete_beta_function (136 words)

	[No title]
		GammaRegularized GammaRegularized[a, z] is the regularized incomplete gamma function Q(a, z).
		Hypergeometric1F1Regularized Hypergeometric1F1Regularized[a, b, z] is the regularized confluent hypergeometric function Hypergeometric1F1[a, b, z]/Gamma[b].
		LogIntegral LogIntegral[z] is the logarithmic integral function li(z).
	www.asergeev.com /files/bgu/math/function/allowed1.txt (723 words)

	The Mathematica Book Online: Advanced Mathematics in Mathematica \| Mathematical Functions
		Notice that in the incomplete beta function, the parameter z is an upper limit of integration, and appears as the first argument of the function.
		The incomplete beta and gamma functions, and their inverses, are common in statistics.
		The zeta function for complex arguments is central to number-theoretical studies of the distribution of primes.
	documents.wolfram.com /mathematica/book/section-3.2.11 (2334 words)

	Beta Function (Site not responding. Last check: 2007-11-05)
		Recycler Beta Function Distortions from Systematic and Random Gradient Errors in...
		Regularized incomplete beta function: Continued fraction representations (formul...
		PlanetMath: evaluation of beta function using Laplace transform...
	www.scienceoxygen.com /math/221.html (142 words)

	Mathematica Usage Messages
		Alias is an obsolete function, superseded by $Pre and $PreRead.
		ColorFunction is an option for various graphics functions which specifies a function to apply to z values to determine the color to use for a particular x, y region.
		Compose is an obsolete function, superseded by Composition.
	vlado.fmf.uni-lj.si /vlado/symbol/usage.htm (11884 words)

	Citations: Pocketbook of Mathematical Functions - Abramowitz, Stegun (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Function erfc Returns the complementary error function erfc(x) 21] Function Simpson Returns the integral of a function which is given at equally spaced abscissas.
		Abramowitz and I. Stegun, Pocketbook of mathematical functions (Verlag Harri Deutsch, Frankfurt/Main, 1984).
		an upper bound for the Bessel functions and, thus, the c k is given by jc k (r)j = j2 J k (r)j 2 minf1; j(r) 2j k (k 1) g k 1: Figure 5 illustrates the decrease of the exact coefficients and our upper bound.
	citeseer.ist.psu.edu /context/478148/0 (1555 words)

	Negative binomial distribution - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-05)
		Third, the negative binomial distribution arises as a continuous mixture of Poisson distributions where the mixing distribution of the Poisson rate is a gamma distribution.
		The cumulative distribution function can be expressed in terms of the regularized incomplete beta function :
		In this case the gamma function is not needed to express the probability mass function, and factorial s or binomial coefficient s can be used instead:
	encyclopedia.worldsearch.com /negative_binomial_distribution.htm (1740 words)

	Gamma Function (Site not responding. Last check: 2007-11-05)
		Paul Godfrey on the Lanczos Implementation of the Gamma Function - Numericana...
		The use of gamma functions in the calculation of organ perfusion functions for n...
		Statistical Distributions - Inverted Beta Distribution - Gamma Function...
	www.scienceoxygen.com /math/222.html (130 words)

	Binomial distribution - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-05)
		However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials.
		The cumulative distribution function can be expressed in terms of the regularized incomplete beta function, as follows:
		Note: that the picture gives the normal and binomial probability density function s (PDF) and not the cumulative distribution function s.
	encyclopedia.worldsearch.com /binomial_distribution.htm (738 words)

	[No title]
		It is actually an identity involving values of the incomplete gamma-function, and if these are given by asymptotic expansions in terms of the complementary error function, the result becomes a formula of the Riemann-Siegel type.
		Whereas the study of the quantum mechanical saddle oscillator by means of periodic orbit theory is an interesting investigation, the similarity of Lorentzian smoothing in the density of states, respectively in the density of zeros for $\zeta(z)$, seems to be coincidental.
		An explicit formula is viewed as establishing the relation between the sum of a suitable test function over the prime powers and the sum of the Mellin transform of the function over the zeros of the zeta function, plus an analytical functional evaluated on the test function.
	www.math.niu.edu /~rusin/known-math/99/zeta (3655 words)

	Gamma and Related Functions - Special Functions - Math Library for C# and VB.NET: BLAS, LAPACK, more
		The Beta function is closely related to the Gamma function.
		The incomplete Beta function and the regularized Beta function have definitions that are analogous to the similarly named Gamma functions.
		The DiGamma function Psi( x) is the derivative of the logarithm of the Gamma function.
	www.extremeoptimization.com /Mathematics/UsersGuide/SpecialFunctions/GammaFunctions.aspx (344 words)

	GammaRelated Members (Site not responding. Last check: 2007-11-05)
		Evaluate the incomplete gamma function defined by GammaI = integral from t = 0 to x of exp(-t) * t**(a-1.0) GammaI(x,a) is evaluated for positive values of a and non-negative values of x.
		GammaIT(x,a) evaluates Tricomi's incomplete gamma function defined by GammaIT = x*(-a)/Gamma(a) integral from 0 to x of exp(-t) * t**(a-1.0) for a > 0.0 and by analytic continuation for a <= 0.0.
		Evaluate the incomplete regularized Gamma function GammaRegularized(a,z) = { integral from z to infinity of exp(-t) * t(a-1) } / Gamma(a) The order of parameters is the same as the Mathematica function** GammaRegularized[a,z].
	altaxo.sourceforge.net /CoreClassRef/Altaxo.Calc.GammaRelatedMembers.html (440 words)

	R: The Regularized Beta Function (sfunc) (Site not responding. Last check: 2007-11-05)
		The Beta value can be scaled to a base 10 logarithm.
		non-negative numeric vectors, the parameters of the regularized Beta function
		The regularized Beta function scales the incomplete Beta function to the interval [0,1], by dividing through B(a,b), i.e.
	www.collocations.de /UCS/UCS-R-html/Rbeta.html (116 words)

	Ra-Rm
		The goal of the project producing R is to demonstrate that it is possible to produce an S-like environment which does not suffer from the memory demands and performance problems of the current S implementation.
		, functions for the bootstrap, cross-validation and the jackknife;
		Functions are also objects and as such can call or be called by other functions, including themselves.
	stommel.tamu.edu /~baum/linuxlist/linuxlist/node41.html (11620 words)

	GammaFunctions.IncompleteBeta Method
		The Incomplete Beta function evaluated for the specified arguments.
		The term 'incomplete' refers to the fact that the defining integral of the beta function is evaluated from zero up to a point between 0 and 1, so that
		It may also refer to the normalized incomplete beta function or regularized beta function, implemented by the RegularizedBeta method.
	www.extremeoptimization.com /Mathematics/Reference/Extreme.Mathematics.SpecialFunctions.GammaFunctions.IncompleteBeta.html (121 words)

	F-distribution - Wikipedia, the free encyclopedia
		The F -distribution arises frequently as the null distribution of a test statistic, especially in likelihood-ratio tests, perhaps most notably in the analysis of variance ; see F-test.
		The probability density function of an F ( d
		where I is the regularized incomplete beta function.
	en.wikipedia.org /wiki/F-distribution (166 words)

	R: Introduction to UCS/R
		Throughout the documentation, it is assumed that you are familiar with the UCS/Perl naming conventions and data set file format.
		This module implements extensions of several association measures to continuous functions on a real-valued coordinate space (generalised association measures, GAMs).
		The functions in this module compute GAM scores and iso-surfaces in standard or ebo-coordinates, and can add jitter to a given data set.
	www.collocations.de /UCS/UCS-R-html (493 words)

	Which Equation Explains the Market Cycle ? best Abel-Regularized Sum (Site not responding. Last check: 2007-11-05)
		function which is the sum of a regular function throughout...
		Incomplete) Beta Function ( Regularized) Beta Prime Distribution beta...
		are derived for the regularized vacuum energy density and radial...
	ascot.pl /th/Fourier1/Abel-Regularized-Sum.htm (443 words)

	GiNaC, an open framework for symbolic computation within the C++ programming language
		All function wrappers are implemented as simple inline functions which just call the corresponding method and are only provided for users uncomfortable with OO who are dead set to avoid method invocations.
		For functions that have a branch cut in the complex plane GiNaC follows the conventions for C++ as defined in the ANSI standard as far as possible.
		The inverse trigonometric and hyperbolic functions are not defined for complex arguments by the C++ standard, however.
	cens.ioc.ee /local/man/GiNaC/tutorial (13219 words)

	Negative binomial distribution (Site not responding. Last check: 2007-11-05)
		: Probability mass function f''(''x) = P( X = x) = probability that r th success occurs on the x th trial ::= C( x − 1, r − 1) p
		: Cumulative distribution function F''(''x) = P( X ≤ x) = probability that r th success occurs on or before the x th trial : No simple closed form solution exists, but this can be computed via the regularized incomplete Beta function as with the binomial distribution.
		Richard Chandler's software Fortran 77 code for random number generation (uniform, exponential, normal, binomial, poisson, geometric, gamma, beta, negative binomial and Weibull), and GLIMCLIM (Generalised Linear Modelling of daily climate sequences).
	www.serebella.com /encyclopedia/article-Negative_binomial_distribution.html (1054 words)

	Citebase - Zero modes, beta functions and IR/UV interplay in higher-loop QED
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
		We analyze the relation between the short-distance behavior of quantum field theory and the strong-field limit of the background field formalism, for QED effective Lagrangians in self-dual backgrounds, at both one and two loop.
		The self-duality of the background leads to zero modes in the case of spinor QED, and these zero modes must be taken into account before comparing the perturbative beta function coefficients and the coefficients of the strong-field limit of the effective Lagrangian.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0210240 (1862 words)

	CodeCogs: Maths::Special::Gamma Namespace Reference - C / C++ Numerical Component (Site not responding. Last check: 2007-11-05)
		Continued fraction expansion #1 for the incomplete beta integral.
		The inverse of the regularized upper incomplete Gamma integral.
		Returns the natural logarithm of the gamma function of the sum of two numbers.
	www.codecogs.com /d-ox/namespace_maths_1_1_special_1_1_gamma.php (106 words)

	negative binomial distribution (Site not responding. Last check: 2007-11-05)
		Probability mass function f ( x) = P( X = x) = probability that r th success occurs on the x th trial
		Cumulative distribution function F ( x) = P( X ≤ x) = probability that r th success occurs on or before the x th trial : No simple closed form solution exists, but this can be computed via the regularized incomplete Beta function as with the binomial distribution.
		What's the probability mass function for selling the last candy bar at the x th house?
	www.yourencyclopedia.net /negative_binomial_distribution.html (853 words)

	Product Features - Math Library for C# and VB.NET: BLAS, LAPACK, more
		Static operator functions for languages that don't support operator overloading.
		Functions from number theory: greatest common divisor, least common multiple, decomposition into prime factors.
		Gamma and related functions, including incomplete and regularized gamma function, digamma function, beta function, harmonic numbers.
	www.extremeoptimization.com /Mathematics/features.aspx (342 words)

	Science Fair Projects - Binomial distribution
		However, the k successes can occur anywhere among the n trials, and there are C( n, k) different ways of distributing k successes in a sequence of n trials.
		Note: that the picture gives the normal and binomial probability density functions (PDF) and not the cumulative distribution functions.
		For example, suppose you randomly sample n people out of a large population and ask them whether they agree with a certain statement.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Binomial_distribution (819 words)

	beta function - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "beta function" is defined.
		Beta Function : Eric Weisstein's World of Mathematics [ home, info ]
		Phrases that include beta function : incomplete beta function, central beta function, complete beta function, dirichlet beta function, regularized beta function
	www.onelook.com /?w=beta+function (125 words)

	Release Notes
		The definitions and notation for incomplete and regularized gamma and beta functions are not consistent, For example, note the differences with the definitions used in Mathematica :
		These function suffixes are mainly intended for future testing of functions and cannot be used in equations.
		The first argument gives the position of the element of the Hessian or gradient element desired in the form i or i:j, where i is the row elemena dn j the column element.
	www.gams.com /docs/release/release.htm (6614 words)

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