org.apache.commons.math3.distribution

## Class PoissonDistribution

- java.lang.Object
- - org.apache.commons.math3.distribution.AbstractIntegerDistribution
  - - org.apache.commons.math3.distribution.PoissonDistribution

All Implemented Interfaces:
  Serializable, IntegerDistribution

    public class PoissonDistribution
    extends AbstractIntegerDistribution

Implementation of the Poisson distribution.

Version:
  $Id: PoissonDistribution.java 1416643 2012-12-03 19:37:14Z tn $
See Also:
  Poisson distribution (Wikipedia), Poisson distribution (MathWorld), Serialized Form

### Field Summary

Fields
`static double` `DEFAULT_EPSILON` Default convergence criterion.
`static int` `DEFAULT_MAX_ITERATIONS` Default maximum number of iterations for cumulative probability calculations.
`private double` `epsilon` Convergence criterion for cumulative probability.
`private ExponentialDistribution` `exponential` Distribution needed for the `sample()` method.
`private int` `maxIterations` Maximum number of iterations for cumulative probability.
`private double` `mean` Mean of the distribution.
`private NormalDistribution` `normal` Distribution used to compute normal approximation.
`private static long` `serialVersionUID` Serializable version identifier.

### Fields inherited from class org.apache.commons.math3.distribution.AbstractIntegerDistribution

`random, randomData`

### Constructor Summary

Constructors
Constructor and Description
`PoissonDistribution(double p)` Creates a new Poisson distribution with specified mean.
`PoissonDistribution(double p, double epsilon)` Creates a new Poisson distribution with the specified mean and convergence criterion.
`PoissonDistribution(double p, double epsilon, int maxIterations)` Creates a new Poisson distribution with specified mean, convergence criterion and maximum number of iterations.
`PoissonDistribution(double p, int maxIterations)` Creates a new Poisson distribution with the specified mean and maximum number of iterations.
`PoissonDistribution(RandomGenerator rng, double p, double epsilon, int maxIterations)` Creates a new Poisson distribution with specified mean, convergence criterion and maximum number of iterations.

### Method Summary

Methods
`double` `cumulativeProbability(int x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
`double` `getMean()` Get the mean for the distribution.
`double` `getNumericalMean()` Use this method to get the numerical value of the mean of this distribution.
`double` `getNumericalVariance()` Use this method to get the numerical value of the variance of this distribution.
`int` `getSupportLowerBound()` Access the lower bound of the support.
`int` `getSupportUpperBound()` Access the upper bound of the support.
`boolean` `isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
`private long` `nextPoisson(double meanPoisson)`
`double` `normalApproximateProbability(int x)` Calculates the Poisson distribution function using a normal approximation.
`double` `probability(int x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X = x)`.
`int` `sample()` Generate a random value sampled from this distribution.

### Methods inherited from class org.apache.commons.math3.distribution.AbstractIntegerDistribution

`cumulativeProbability, inverseCumulativeProbability, reseedRandomGenerator, sample, solveInverseCumulativeProbability`

### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

### Field Detail

#### DEFAULT_MAX_ITERATIONS

    public static final int DEFAULT_MAX_ITERATIONS

Default maximum number of iterations for cumulative probability calculations.

Since:
  2.1
See Also:
  Constant Field Values

#### DEFAULT_EPSILON

    public static final double DEFAULT_EPSILON

Default convergence criterion.

Since:
  2.1
See Also:
  Constant Field Values

#### serialVersionUID

    private static final long serialVersionUID

Serializable version identifier.

See Also:
  Constant Field Values

#### normal

    private final NormalDistribution normal

Distribution used to compute normal approximation.

#### exponential

    private final ExponentialDistribution exponential

Distribution needed for the `sample()` method.

#### mean

    private final double mean

Mean of the distribution.

#### maxIterations

    private final int maxIterations

Maximum number of iterations for cumulative probability. Cumulative probabilities are estimated using either Lanczos series approximation of `Gamma.regularizedGammaP(double, double, double, int)` or continued fraction approximation of `Gamma.regularizedGammaQ(double, double, double, int)`.

#### epsilon

    private final double epsilon

Convergence criterion for cumulative probability.

### Constructor Detail

#### PoissonDistribution

    public PoissonDistribution(double p)
                        throws NotStrictlyPositiveException

Creates a new Poisson distribution with specified mean.

Parameters:
  `p` - the Poisson mean
Throws:
  `NotStrictlyPositiveException` - if `p <= 0`.

#### PoissonDistribution

    public PoissonDistribution(double p,
                       double epsilon,
                       int maxIterations)
                        throws NotStrictlyPositiveException

Creates a new Poisson distribution with specified mean, convergence criterion and maximum number of iterations.

Parameters:
  `p` - Poisson mean.
  `epsilon` - Convergence criterion for cumulative probabilities.
  `maxIterations` - the maximum number of iterations for cumulative probabilities.
Throws:
  `NotStrictlyPositiveException` - if `p <= 0`.
Since:
  2.1

#### PoissonDistribution

    public PoissonDistribution(RandomGenerator rng,
                       double p,
                       double epsilon,
                       int maxIterations)
                        throws NotStrictlyPositiveException

Creates a new Poisson distribution with specified mean, convergence criterion and maximum number of iterations.

Parameters:
  `rng` - Random number generator.
  `p` - Poisson mean.
  `epsilon` - Convergence criterion for cumulative probabilities.
  `maxIterations` - the maximum number of iterations for cumulative probabilities.
Throws:
  `NotStrictlyPositiveException` - if `p <= 0`.
Since:
  3.1

#### PoissonDistribution

    public PoissonDistribution(double p,
                       double epsilon)
                        throws NotStrictlyPositiveException

Creates a new Poisson distribution with the specified mean and convergence criterion.

Parameters:
  `p` - Poisson mean.
  `epsilon` - Convergence criterion for cumulative probabilities.
Throws:
  `NotStrictlyPositiveException` - if `p <= 0`.
Since:
  2.1

#### PoissonDistribution

    public PoissonDistribution(double p,
                       int maxIterations)

Creates a new Poisson distribution with the specified mean and maximum number of iterations.

Parameters:
  `p` - Poisson mean.
  `maxIterations` - Maximum number of iterations for cumulative probabilities.
Since:
  2.1

### Method Detail

#### getMean

    public double getMean()

Get the mean for the distribution.

Returns:
  the mean for the distribution.

#### 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` - the point at which the PMF is evaluated
Returns:
  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` - the 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`

#### normalApproximateProbability

    public double normalApproximateProbability(int x)

Calculates the Poisson distribution function using a normal approximation. The `N(mean, sqrt(mean))` distribution is used to approximate the Poisson distribution. The computation uses "half-correction" (evaluating the normal distribution function at `x + 0.5`).

Parameters:
  `x` - Upper bound, inclusive.
Returns:
  the distribution function value calculated using a normal approximation.

#### getNumericalMean

    public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution. For mean parameter `p`, the mean is `p`.

Returns:
  the mean or `Double.NaN` if it is not defined

#### getNumericalVariance

    public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution. For mean parameter `p`, the variance is `p`.

Returns:
  the variance (possibly `Double.POSITIVE_INFINITY` or `Double.NaN` if it is not defined)

#### getSupportLowerBound

    public int getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as `inverseCumulativeProbability(0)`. In other words, this method must return

`inf {x in Z | P(X <= x) > 0}`.

The lower bound of the support is always 0 no matter the mean parameter.

Returns:
  lower bound of the support (always 0)

#### getSupportUpperBound

    public int getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as `inverseCumulativeProbability(1)`. In other words, this method must return

`inf {x in R | P(X <= x) = 1}`.

The upper bound of the support is positive infinity, regardless of the parameter values. There is no integer infinity, so this method returns `Integer.MAX_VALUE`.

Returns:
  upper bound of the support (always `Integer.MAX_VALUE` for positive infinity)

#### isSupportConnected

    public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all integers between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:
  `true`

#### sample

    public int sample()

Generate a random value sampled from this distribution. The default implementation uses the inversion method.

Algorithm Description:

- For small means, uses simulation of a Poisson process using Uniform deviates, as described here. The Poisson process (and hence value returned) is bounded by 1000 * mean.
- For large means, uses the rejection algorithm described in <quote> Devroye, Luc. (1981).The Computer Generation of Poisson Random Variables Computing vol. 26 pp. 197-207. </quote>

Specified by:
  `sample` in interface `IntegerDistribution`
Overrides:
  `sample` in class `AbstractIntegerDistribution`
Returns:
  a random value.
Since:
  2.2

#### nextPoisson

    private long nextPoisson(double meanPoisson)

Parameters:
  `meanPoisson` - Mean of the Poisson distribution.
Returns:
  the next sample.