gives the regularized incomplete beta function .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For nonsingular cases, .
- BetaRegularized[z0,z1,a,b] gives the generalized regularized incomplete beta function defined in nonsingular cases as Beta[z0,z1,a,b]/Beta[a,b].
- Note that the arguments in BetaRegularized are arranged differently from those in GammaRegularized.
- For certain special arguments, BetaRegularized automatically evaluates to exact values.
- BetaRegularized can be evaluated to arbitrary numerical precision.
- BetaRegularized automatically threads over lists.
- BetaRegularized can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Asymptotic expansion at Infinity:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
BetaRegularized can be used with Interval and CenteredInterval objects:
Specific Values (4)
Values of BetaRegularized at fixed points:
Find a value of z for which the BetaRegularized[z,1,3]=3.5:
Plot the BetaRegularized function for various parameters:
Function Properties (9)
is defined for all real and complex values:
The regularized incomplete beta function is an analytic function of for positive integer :
Thus, any such function will have no singularities or discontinuities:
For other values of , is neither analytic nor meromorphic:
is neither non-increasing nor non-decreasing:
is injective for positive odd but not positive even :
is surjective for positive odd but not positive even :
is non-negative for positive even but indefinite for odd :
Compute the indefinite integral using Integrate:
Series Expansions (5)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Find the series expansion at Infinity:
Find the series expansion for an arbitrary symbolic direction :
Function Identities and Simplifications (3)
Regularized incomplete beta function is related to the incomplete beta function:
BetaRegularized may reduce to a simpler form:
Generalizations & Extensions (8)
Ordinary Regularized Incomplete Beta Function (5)
Evaluate at integer and half‐integer arguments:
Infinite arguments give symbolic results:
BetaRegularized threads elementwise over lists:
BetaRegularized can be applied to power series:
Series expansion at infinity:
Give the result for an arbitrary symbolic direction:
Plot of the absolute value of BetaRegularized in the complex plane:
Distribution of the average distance s of all pairs of points in a d‐dimensional hypersphere:
Low‐dimensional distributions can be expressed in elementary functions:
CDF of the Student t distribution:
Plot the CDF for various parameters:
The CDF of the StudentTDistribution is given in terms of BetaRegularized functions:
Properties & Relations (3)
Use FunctionExpand to express through Gamma and Beta functions:
Numerically find a root of a transcendental equation:
Compose with the inverse function:
Use PowerExpand to disregard multivaluedness ambiguity:
Possible Issues (3)
Large arguments can give results too large to be computed explicitly:
Machine‐number inputs can give high‐precision results:
Regularized beta functions are typically not generated by FullSimplify:
Wolfram Research (1991), BetaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaRegularized.html (updated 2022).
Wolfram Language. 1991. "BetaRegularized." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/BetaRegularized.html.
Wolfram Language. (1991). BetaRegularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaRegularized.html