umontreal.iro.lecuyer.probdist
Class BetaDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.BetaDist

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

BetaSymmetricalDist

public class BetaDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the beta distribution with shape parameters α > 0 and β > 0, over the interval (a, b), where a < b. It has density

f (x) = (x - a)^α-1(b - x)^β-1/[B(α, β)(b - a)^α+β-1]

for a < x < b, and 0 elsewhere. It has distribution function

F(x) = I_{α, β}(x) = ∫_a^x(ξ - a)^α-1(b - ξ)^β-1/[B(α, β)(b - a)^α+β-1]dξ, for a < x < b, and 0 elsewhere,

where B(α, β) is the beta function defined by

B(α, β) = Γ(α)Γ(β)/Γ(α + β)

and Γ(x) is the gamma function defined in GammaDist.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

BetaDist(double alpha, double beta)
Constructs a BetaDist object with parameters α = alpha and β = beta and default domain (0, 1).

BetaDist(double alpha, double beta, double a, double b)
Constructs a BetaDist object with parameters α = alpha and β = beta, and domain (a, b).

BetaDist(double alpha, double beta, double a, double b, int d)
Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing distribution, complementary distribution, and inverse functions.

BetaDist(double alpha, double beta, int d)
Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.

Method Summary

static double	barF(double alpha, double beta, double a, double b, int d, double x) Computes the complementary distribution function.
static double	barF(double alpha, double beta, int d, double x) Same as barF (alpha, beta, 0, 1, d, x).
double	cdf(double x) Computes and returns the distribution function F(x).
static double	cdf(double alpha, double beta, double a, double b, int d, double x) Computes an approximation of the distribution function, with roughly d decimal digits of precision.
static double	cdf(double alpha, double beta, int d, double x) Same as cdf (alpha, beta, 0, 1, d, x).
double	density(double x) Returns f (x), the density of X evaluated at x.
static double	density(double alpha, double beta, double x) Same as density (alpha, beta, 0, 1, x).
static double	density(double alpha, double beta, double a, double b, double x) Computes the density function of the beta distribution.
double	getA() Returns the parameter a of this object.
double	getAlpha() Returns the parameter α of this object.
double	getB() Returns the parameter b of this object.
double	getBeta() Returns the parameter β of this object.
double	inverseF(double u) Computes and returns the inverse distribution function F-1(u), defined in.
static double	inverseF(double alpha, double beta, double a, double b, int d, double u) Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library.
static double	inverseF(double alpha, double beta, int d, double u) Same as inverseF (alpha, beta, 0, 1, d, u).
void	setParams(double alpha, double beta, double a, double b, int d)

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

barF, inverseBisection, inverseBrent

Methods inherited from class java.lang.Object

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

Constructor Detail

BetaDist

public BetaDist(double alpha,
                double beta)

Constructs a BetaDist object with parameters α = alpha and β = beta and default domain (0, 1).

BetaDist

public BetaDist(double alpha,
                double beta,
                double a,
                double b)

Constructs a BetaDist object with parameters α = alpha and β = beta, and domain (a, b).

BetaDist

public BetaDist(double alpha,
                double beta,
                int d)

Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions. The default domain (0, 1) is used.

BetaDist

public BetaDist(double alpha,
                double beta,
                double a,
                double b,
                int d)

Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing distribution, complementary distribution, and inverse functions. The domain (a, b) is used.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density of X evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Computes and returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from interface: Distribution

Computes and returns the inverse distribution function F^-1(u), defined in.

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value in the interval (0, 1) for which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

density

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

Same as density (alpha, beta, 0, 1, x).

density

public static double density(double alpha,
                             double beta,
                             double a,
                             double b,
                             double x)

Computes the density function of the beta distribution.

cdf

public static double cdf(double alpha,
                         double beta,
                         int d,
                         double x)

Same as cdf (alpha, beta, 0, 1, d, x).

cdf

public static double cdf(double alpha,
                         double beta,
                         double a,
                         double b,
                         int d,
                         double x)

Computes an approximation of the distribution function, with roughly d decimal digits of precision.

barF

public static double barF(double alpha,
                          double beta,
                          int d,
                          double x)

Same as barF (alpha, beta, 0, 1, d, x).

barF

public static double barF(double alpha,
                          double beta,
                          double a,
                          double b,
                          int d,
                          double x)

Computes the complementary distribution function.

inverseF

public static double inverseF(double alpha,
                              double beta,
                              int d,
                              double u)

Same as inverseF (alpha, beta, 0, 1, d, u).

inverseF

public static double inverseF(double alpha,
                              double beta,
                              double a,
                              double b,
                              int d,
                              double u)

Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library. The method performs interval halving or Newton iterations to compute the inverse. The precision depends on the accuracy of the cdf method. The argument d gives a good idea of the precision attained.

getAlpha

public double getAlpha()

Returns the parameter α of this object.

getBeta

public double getBeta()

Returns the parameter β of this object.

getA

public double getA()

Returns the parameter a of this object.

getB

public double getB()

Returns the parameter b of this object.

setParams

public void setParams(double alpha,
                      double beta,
                      double a,
                      double b,
                      int d)