Package org.apache.commons.statistics.distribution

Class ExponentialDistribution

  • All Implemented Interfaces:
    ContinuousDistribution
    public final class ExponentialDistribution
extends AbstractContinuousDistribution
    Implementation of the exponential distribution.

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

    \[ f(x; \mu) = \frac{1}{\mu} e^{-x / \mu} \]

    for \( \mu > 0 \) the mean and \( x \in [0, \infty) \).

    This implementation uses the scale parameter \( \mu \) which is the mean of the distribution. A common alternative parameterization uses the rate parameter \( \lambda \) which is the reciprocal of the mean. The distribution can be be created using \( \mu = \frac{1}{\lambda} \).

    See Also:
    Exponential distribution (Wikipedia), Exponential distribution (MathWorld)

    • Field Summary

      Fields 
      Modifier and Type Field Description
      private double logMean
      The logarithm of the mean, stored to reduce computing time.
      private double mean
      The mean of this distribution.
      private static double SUPPORT_HI
      Support upper bound.
      private static double SUPPORT_LO
      Support lower bound.

    • Constructor Summary

      Constructors 
      Modifier Constructor Description
      private ExponentialDistribution​(double mean)  

    • Field Detail

      • SUPPORT_LO

        private static final double SUPPORT_LO
        Support lower bound.
        See Also:
        Constant Field Values

      • SUPPORT_HI

        private static final double SUPPORT_HI
        Support upper bound.
        See Also:
        Constant Field Values

      • mean

        private final double mean
        The mean of this distribution.

      • logMean

        private final double logMean
        The logarithm of the mean, stored to reduce computing time.

    • Constructor Detail

      • ExponentialDistribution

        private ExponentialDistribution​(double mean)
        Parameters:
        mean - Mean of this distribution.

    • Method Detail

      • of

        public static ExponentialDistribution of​(double mean)
        Creates an exponential distribution.
        Parameters:
        mean - Mean of this distribution. This is a scale parameter.
        Returns:
        the distribution
        Throws:
        java.lang.IllegalArgumentException - if mean <= 0.

      • 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.
        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.
        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.

      • 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:

        Returns 0 when p == 1 and Double.POSITIVE_INFINITY when p == 0.

        Specified by:
        inverseSurvivalProbability in interface ContinuousDistribution
        Overrides:
        inverseSurvivalProbability in class AbstractContinuousDistribution
        Parameters:
        p - Survival probability.
        Returns:
        the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

      • getMean

        public double getMean()
        Gets the mean of this distribution.
        Returns:
        the mean.

      • getVariance

        public double getVariance()
        Gets the variance of this distribution.

        For mean \( \mu \), the variance is \( \mu^2 \).

        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 positive infinity.

        Returns:
        positive infinity.