Class BetaDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.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:

  • Field Details

    • 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 Details

    • BetaDistribution

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

  • 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
      Overrides:
      createSampler in class AbstractContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.