## 8 Probability Distributions

### 8.1 Overview

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

An overview of available continuous distributions:

### 8.2 Distribution Framework

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.

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).

TDistribution t = new TDistribution(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 inverseCumulativeProbability methods. 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.

### 8.3 User Defined Distributions

Since there are numerous distributions and Commons-Math only directly supports a handful, it may be necessary to extend the distribution framework to satisfy individual needs. It is recommended that the RealDistribution, IntegerDistribution and MultivariateRealDistribution interfaces serve as base types for any extension. These serve as the basis for all the distributions directly supported by Commons-Math and using those interfaces for implementation purposes will ensure any extension is compatible with the remainder of Commons-Math. To aid in implementing a distribution extension, the AbstractRealDistribution, AbstractIntegerDistribution and AbstractMultivariateRealDistribution provide implementation building blocks and offer basic distribution functionality. By extending these abstract classes directly, much of the repetitive distribution implementation is already developed and should save time and effort in developing user-defined distributions.