Package org.apache.commons.statistics.distribution

Class PoissonDistribution

java.lang.Object
org.apache.commons.statistics.distribution.PoissonDistribution
All Implemented Interfaces:
DiscreteDistribution
public final class PoissonDistribution extends Object
Implementation of the Poisson distribution.

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

\[ f(k; \lambda) = \frac{\lambda^k e^{-k}}{k!} \]

for \( \lambda \in (0, \infty) \) the mean and \( k \in \{0, 1, 2, \dots\} \) the number of events.

See Also:

  • Method Details

    • of

      public static PoissonDistribution of(double mean)
      Creates a Poisson distribution.
      Parameters:
      mean - Poisson mean.
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if mean <= 0.

    • probability

      public double probability(int 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 probability mass function (PMF) for the distribution.
      Parameters:
      x - Point at which the PMF is evaluated.
      Returns:
      the value of the probability mass function at x.

    • logProbability

      public double logProbability(int x)
      For a random variable X whose values are distributed according to this distribution, this method returns log(P(X = x)), where log is the natural logarithm.
      Parameters:
      x - Point at which the PMF is evaluated.
      Returns:
      the logarithm of the value of the probability mass function at x.

    • cumulativeProbability

      public double cumulativeProbability(int 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(int 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.
      Returns:
      the mean.

    • getVariance

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

      The variance is equal to the mean.

      Returns:
      the variance.

    • getSupportLowerBound

      public int getSupportLowerBound()
      Gets the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) \gt 0 \} \). By convention, Integer.MIN_VALUE should be substituted for negative infinity.

      The lower bound of the support is always 0.

      Returns:
      0.

    • getSupportUpperBound

      public int getSupportUpperBound()
      Gets the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) = 1 \} \). By convention, Integer.MAX_VALUE should be substituted for positive infinity.

      The upper bound of the support is always positive infinity.

      Returns:
      Integer.MAX_VALUE

    • createSampler

      public DiscreteDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Creates a sampler.
      Specified by:
      createSampler in interface DiscreteDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.

    • probability

      public double probability(int x0, int 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)

      Special cases:

      • returns 0.0 if x0 == x1;
      • returns probability(x1) if x0 + 1 == x1;
      Specified by:
      probability in interface DiscreteDistribution
      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 int inverseCumulativeProbability(double p)
      Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb Z : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb Z : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]

      If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of cumulativeProbability(x) called using the returned p-quantile may not compute the original p.

      The default implementation returns:

      Specified by:
      inverseCumulativeProbability in interface DiscreteDistribution
      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 int 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:

      \[ x = \begin{cases} \inf \{ x \in \mathbb Z : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb Z : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

      If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of survivalProbability(x) called using the returned (1-p)-quantile may not compute the original p.

      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 DiscreteDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).
      Throws:
      IllegalArgumentException - if p < 0 or p > 1