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| <?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?> |
| <document url="distribution.html"> |
| <properties> |
| <title>The Commons Math User Guide - Distributions</title> |
| </properties> |
| <body> |
| <section name="1 Probability Distributions"> |
| <subsection name="1.1 Overview" href="overview"> |
| <p> |
| The distributions package provides a framework and implementations for some commonly used |
| probability distributions. Continuous univariate distributions are represented by implementations of |
| the <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/ContinuousDistribution.html">ContinuousDistribution</a> |
| interface. Discrete distributions implement |
| <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/DiscreteDistribution.html">DiscreteDistribution</a> |
| (values must be mapped to integers). |
| </p> |
| </subsection> |
| <subsection name="1.2 API" href="distributions"> |
| <p> |
| The distribution framework provides the means to compute probability density, |
| probability mass and cumulative probability functions for several well-known |
| discrete (integer-valued) and continuous probability distributions. |
| The API also allows for the computation of inverse cumulative probabilities |
| and sampling from distributions. |
| </p> |
| <p> |
| For an instance <code>f</code> of a distribution <code>F</code>, |
| and a domain value, <code>x</code>, <code>f.cumulativeProbability(x)</code> |
| computes <code>P(X <= x)</code> where <code>X</code> is a random variable distributed |
| as <code>F</code>. |
| For <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/DiscreteDistribution.html">discrete</a> |
| <code>F</code>, the probability mass function is given by <code>f.probability(x)</code>. |
| For <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/ContinuousDistribution.html">continuous</a> |
| <code>F</code>, the probability density function is given by <code>f.density(x)</code>. |
| Continuous distributions also implement <code>f.probability(x1, x2)</code> for computing |
| <code>P(x1 <= X <= x2)</code>. |
| </p> |
| <source>TDistribution t = new TDistribution(29); |
| double lowerTail = t.cumulativeProbability(-2.656); // P(T(29) <= -2.656) |
| double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T(29) >= 2.75)</source> |
| <p> |
| All distributions implement a <code>createSampler(UniformRandomProvider rng)</code> |
| method to support random sampling from the distribution, where <code>UniformRandomProvider</code> |
| is an interface defined in <a href="https://commons.apache.org/rng">Commons RNG</a>. |
| </p> |
| <p> |
| Inverse distribution functions can be computed using the |
| <code>inverseCumulativeProbability</code> methods. For continuous <code>f</code> |
| and <code>p</code> a probability, <code>f.inverseCumulativeProbability(p)</code> returns |
| <ul> |
| <li><code>inf{x in R | P(X ≤ x) ≥ p} for 0 < p < 1},</code></li> |
| <li><code>inf{x in R | P(X ≤ x) > 0} for p = 0}.</code></li> |
| </ul> |
| where <code>X</code> is distributed as <code>F</code>.<br/> |
| For discrete <code>F</code>, the definition is the same, with <code>Z</code> (the integers) |
| in place of <code>R</code> (but note that, in the discrete case, the ≥ in the definition |
| can make a difference when <code>p</code> is an attained value of the distribution). |
| </p> |
| </subsection> |
| </section> |
| </body> |
| </document> |
