java.lang.Object

org.apache.commons.statistics.distribution.BetaDistribution

All Implemented Interfaces:: ContinuousDistribution

public final class BetaDistribution extends Object

Implementation of the beta distribution.

The probability density function of \( X \) is:

\[ f(x; \alpha, \beta) = \frac{1}{ B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1} \]

for \( \alpha > 0 \), \( \beta > 0 \), \( x \in [0, 1] \), and the beta function, \( B \), is a normalization constant:

\[ B(\alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} \]

where \( \Gamma \) is the gamma function.

\( \alpha \) and \( \beta \) are shape parameters.

See Also:

Nested Class Summary

Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
Method Summary

Modifier and Type

Method

Description

ContinuousDistribution.Sampler

createSampler(org.apache.commons.rng.UniformRandomProvider rng)

Creates a sampler.

double

cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).

double

density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x.

double

getAlpha()

Gets the first shape parameter of this distribution.

double

getBeta()

Gets the second shape parameter of this distribution.

double

getMean()

Gets the mean of this distribution.

double

getSupportLowerBound()

Gets the lower bound of the support.

double

getSupportUpperBound()

Gets the upper bound of the support.

double

getVariance()

Gets the variance of this distribution.

double

inverseCumulativeProbability(double p)

Computes the quantile function of this distribution.

double

inverseSurvivalProbability(double p)

Computes the inverse survival probability function of this distribution.

double

logDensity(double x)

Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.

static BetaDistribution

of(double alpha, double beta)

Creates a beta distribution.

double

probability(double x0, double x1)

For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).

double

survivalProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details
- of
  
  public static BetaDistribution of(double alpha, double beta)
  
  Creates a beta distribution.
  
  Parameters:
  
  alpha - First shape parameter (must be positive).
  
  beta - Second shape parameter (must be positive).
  
  Returns:
  
  the distribution
  
  Throws:
  
  IllegalArgumentException - if alpha <= 0 or beta <= 0.
- getAlpha
  
  public double getAlpha()
  
  Gets the first shape parameter of this distribution.
  
  Returns:
  
  the first shape parameter.
- getBeta
  
  public double getBeta()
  
  Gets the second shape parameter of this distribution.
  
  Returns:
  
  the second shape parameter.
- density
  
  public double density(double x)
  
  Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.
  The density is not defined when x = 0, alpha < 1, or x = 1, beta < 1. In this case the limit of infinity is returned.
  
  Parameters:
  
  x - Point at which the PDF is evaluated.
  
  Returns:
  
  the value of the probability density function at x.
- logDensity
  
  public double logDensity(double x)
  
  Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
  The density is not defined when x = 0, alpha < 1, or x = 1, beta < 1. In this case the limit of infinity is returned.
  
  Parameters:
  
  x - Point at which the PDF is evaluated.
  
  Returns:
  
  the logarithm of the value of the probability density function at x.
- cumulativeProbability
  
  public double cumulativeProbability(double x)
  
  For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.
  
  Parameters:
  
  x - Point at which the CDF is evaluated.
  
  Returns:
  
  the probability that a random variable with this distribution takes a value less than or equal to x.
- survivalProbability
  
  public double survivalProbability(double x)
  
  For a random variable X whose values are distributed according to this distribution, this method returns P(X > x). In other words, this method represents the complementary cumulative distribution function.
  By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.
  
  Parameters:
  
  x - Point at which the survival function is evaluated.
  
  Returns:
  
  the probability that a random variable with this distribution takes a value greater than x.
- getMean
  
  public double getMean()
  
  Gets the mean of this distribution.
  For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the mean is:
  \[ \frac{\alpha}{\alpha + \beta} \]
  
  Returns:
  
  the mean.
- getVariance
  
  public double getVariance()
  
  Gets the variance of this distribution.
  For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the variance is:
  \[ \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]
  
  Returns:
  
  the variance.
- getSupportLowerBound
  
  public double getSupportLowerBound()
  
  Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).
  The lower bound of the support is always 0.
  
  Returns:
  
  0.
- getSupportUpperBound
  
  public double getSupportUpperBound()
  
  Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).
  The upper bound of the support is always 1.
  
  Returns:
  
  1.
- createSampler
  
  public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
  
  Creates a sampler.
  
  Specified by:
  
  createSampler in interface ContinuousDistribution
  
  Parameters:
  
  rng - Generator of uniformly distributed numbers.
  
  Returns:
  
  a sampler that produces random numbers according this distribution.
- probability
  
  public double probability(double x0, double x1)
  
  For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
  
  Specified by:
  
  probability in interface ContinuousDistribution
  
  Parameters:
  
  x0 - Lower bound (exclusive).
  
  x1 - Upper bound (inclusive).
  
  Returns:
  
  the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.
- inverseCumulativeProbability
  
  public double inverseCumulativeProbability(double p)
  Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:
  \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
  The default implementation returns:
  
  ContinuousDistribution.getSupportLowerBound() for p = 0,
  
  ContinuousDistribution.getSupportUpperBound() for p = 1, or
  
  the result of a search for a root between the lower and upper bound using cumulativeProbability(x) - p. The bounds may be bracketed for efficiency.
  Specified by:
  
  inverseCumulativeProbability in interface ContinuousDistribution
  
  Parameters:
  
  p - Cumulative probability.
  
  Returns:
  
  the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
  
  Throws:
  
  IllegalArgumentException - if p < 0 or p > 1
- inverseSurvivalProbability
  
  public double inverseSurvivalProbability(double p)
  Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:
  \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
  By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.
  The default implementation returns:
  
  ContinuousDistribution.getSupportLowerBound() for p = 1,
  
  ContinuousDistribution.getSupportUpperBound() for p = 0, or
  
  the result of a search for a root between the lower and upper bound using survivalProbability(x) - p. The bounds may be bracketed for efficiency.
  Specified by:
  
  inverseSurvivalProbability in interface ContinuousDistribution
  
  Parameters:
  
  p - Survival probability.
  
  Returns:
  
  the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).
  
  Throws:
  
  IllegalArgumentException - if p < 0 or p > 1

Class BetaDistribution

Nested Class Summary

Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

Method Summary

Methods inherited from class java.lang.Object

Method Details

of

getAlpha

getBeta

density

logDensity

cumulativeProbability

survivalProbability

getMean

getVariance

getSupportLowerBound

getSupportUpperBound

createSampler

probability

inverseCumulativeProbability

inverseSurvivalProbability