Package smile.math.special

Class Beta

java.lang.Object

smile.math.special.Beta

public class Beta
extends Object

The beta function, also called the Euler integral of the first kind.

B(x, y) = ∫₀¹ t^x-1 (1-t)^y-1dt

for x, y > 0 and the integration is over [0, 1]. The beta function is symmetric, i.e. B(x, y) = B(y, x).

Method Summary

Modifier and Type	Method	Description
static double	beta(double x, double y)	Beta function, also called the Euler integral of the first kind.
static double	inverseRegularizedIncompleteBetaFunction(double alpha, double beta, double p)	Inverse of regularized incomplete beta function.
static double	regularizedIncompleteBetaFunction(double alpha, double beta, double x)	Regularized Incomplete Beta function.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

beta

public static double beta(double x,
 double y)

Beta function, also called the Euler integral of the first kind. The beta function is symmetric, i.e. B(x,y)==B(y,x).

Parameters:

x - x > 0

y - y > 0

Returns:

the function value.

regularizedIncompleteBetaFunction

public static double regularizedIncompleteBetaFunction(double alpha,
 double beta,
 double x)

Regularized Incomplete Beta function. The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function of a random variable X following a binomial distribution.

The implementation employs the continued fraction approximation (see Numerical Recipes).

Parameters:

alpha - alpha > 0

beta - beta > 0

x - x > 0

Returns:

the function value.

inverseRegularizedIncompleteBetaFunction

public static double inverseRegularizedIncompleteBetaFunction(double alpha,
 double beta,
 double p)

Inverse of regularized incomplete beta function.

Parameters:

alpha - alpha > 0

beta - beta > 0

p - 0 <= p <= 1

Returns:

the function value.