Package org.apache.commons.statistics.distribution

Class BetaDistribution

  • All Implemented Interfaces:
    ContinuousDistribution
    public final class BetaDistribution
extends AbstractContinuousDistribution
    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:
    Beta distribution (Wikipedia), Beta distribution (MathWorld)

    • Field Summary

      Fields 
      Modifier and Type Field Description
      private double alpha
      First shape parameter.
      private double beta
      Second shape parameter.
      private double logBeta
      Normalizing factor used in log density computations.
      private double mean
      Cached value for inverse probability function.
      private double variance
      Cached value for inverse probability function.

    • Constructor Summary

      Constructors 
      Modifier Constructor Description
      private BetaDistribution​(double alpha, double beta)  

    • Field Detail

      • alpha

        private final double alpha
        First shape parameter.

      • beta

        private final double beta
        Second shape parameter.

      • logBeta

        private final double logBeta
        Normalizing factor used in log density computations. log(beta(a, b)).

      • mean

        private final double mean
        Cached value for inverse probability function.

      • variance

        private final double variance
        Cached value for inverse probability function.

    • Constructor Detail

      • BetaDistribution

        private BetaDistribution​(double alpha,
                         double beta)
        Parameters:
        alpha - First shape parameter (must be positive).
        beta - Second shape parameter (must be positive).

    • Method Detail

      • 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:
        java.lang.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.