Package org.apache.commons.statistics.distribution

Class 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:
    Beta distribution (Wikipedia), Beta distribution (MathWorld)

    • 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:
        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.

      • 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:

        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:

        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