The Commons Math User Guide

The distributions package provide a framework for some commonly used probability distributions.

The distribution framework provides the means to compute probability density function (PDF) probabilities and cumulative distribution function (CDF) probabilities for common probability distributions. Along with the direct computation of PDF and CDF probabilities, the framework also allows for the computation of inverse PDF and inverse CDF values.

In order to use the distribution framework, first a distribution object must be created. It is encouraged that all distribution object creation occurs via the org.apache.commons.math.distribution.DistributionFactory class. DistributionFactory is a simple factory used to create all of the distribution objects supported by Commons-Math. The typical usage of DistributionFactory to create a distribution object would be:

DistributionFactory factory = DistributionFactory.newInstance(); BinomialDistribution binomial = factory.createBinomialDistribution(10, .75);

The distributions that can be instantiated via the DistributionFactory are detailed below:

Distribution Factory Method Parameters

Binomial createBinomialDistribution
Number of trials
Probability of success

Chi-Squared createChiSquaredDistribution
Degrees of freedom

Exponential createExponentialDistribution
Mean

F createFDistribution
Numerator degrees of freedom
Denominator degrees of freedom

Gamma createGammaDistribution
Alpha
Beta

Hypergeometric createHypogeometricDistribution
Population size
Number of successes in population
Sample size

Normal (Gaussian) createNormalDistribution
Mean
Standard Deviation

t createTDistribution
Degrees of freedom

Distribution	Factory Method	Parameters
Binomial	createBinomialDistribution	Number of trials Probability of success
Chi-Squared	createChiSquaredDistribution	Degrees of freedom
Exponential	createExponentialDistribution	Mean
F	createFDistribution	Numerator degrees of freedom Denominator degrees of freedom
Gamma	createGammaDistribution	Alpha Beta
Hypergeometric	createHypogeometricDistribution	Population size Number of successes in population Sample size
Normal (Gaussian)	createNormalDistribution	Mean Standard Deviation
t	createTDistribution	Degrees of freedom

Using a distribution object, PDF and CDF probabilities are easily computed using the cumulativeProbability methods. For a distribution X, and a domain value, x, cumulativeProbability computes P(X <= x) (i.e. the lower tail probability of X).

DistributionFactory factory = DistributionFactory.newInstance(); TDistribution t = factory.createBinomialDistribution(29); double lowerTail = t.cumulativeProbability(-2.656); // P(T <= -2.656) double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)

The inverse PDF and CDF values are just as easily computed using the inverseCumulativeProbabilitymethods. For a distribution X, and a probability, p, inverseCumulativeProbability computes the domain value x, such that:

P(X <= x) = p, for continuous distributions
P(X <= x) <= p, for discrete distributions

Notice the different cases for continuous and discrete distributions. This is the result of PDFs not being invertible functions. As such, for discrete distributions, an exact domain value can not be returned. Only the "best" domain value. For Commons-Math, the "best" domain value is determined by the largest domain value whose cumulative probability is less-than or equal to the given probability.