org.apache.commons.math3.distribution

## Class PoissonDistribution

• ### Field Summary

Fields
Modifier and Type Field and Description
static double DEFAULT_EPSILON
Default convergence criterion.
static int DEFAULT_MAX_ITERATIONS
Default maximum number of iterations for cumulative probability calculations.
• ### 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
Modifier and Type Method and Description
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.
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.
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
Constant Field Values
• #### DEFAULT_EPSILON

public static final double DEFAULT_EPSILON
Default convergence criterion.
Since:
2.1
Constant Field Values
• ### Constructor Detail

• #### PoissonDistribution

public PoissonDistribution(double p)
throws NotStrictlyPositiveException
Creates a new Poisson distribution with specified mean.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractIntegerDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.

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.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractIntegerDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.

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
• #### 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. In other words, this method represents the logarithm of the probability mass function (PMF) for the distribution. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of IntegerDistribution.probability(int).

The default implementation simply computes the logarithm of probability(x).

Overrides:
logProbability in class AbstractIntegerDistribution
Parameters:
x - the 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 - 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
Devroye, Luc. (1981).The Computer Generation of Poisson Random Variables
Computing vol. 26 pp. 197-207.

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