MATH-1603: Userguide update.
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Gilles Sadowski
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<section name="8 Probability Distributions">
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<subsection name="8.1 Overview" href="overview">
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<p>
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The distributions package provides a framework and implementations for some commonly used
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probability distributions. Continuous univariate distributions are represented by implementations of
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the <a href="../apidocs/org/apache/commons/math4/distribution/RealDistribution.html">RealDistribution</a>
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interface. Discrete distributions implement
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<a href="../apidocs/org/apache/commons/math4/distribution/IntegerDistribution.html">IntegerDistribution</a>
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(values must be mapped to integers) and there is an
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<a href="../apidocs/org/apache/commons/math4/distribution/EnumeratedDistribution.html">EnumeratedDistribution</a>
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class representing discrete distributions with a finite, enumerated set of values. Finally, multivariate
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real-valued distributions can be represented via the
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<a href="../apidocs/org/apache/commons/math4/distribution/MultiVariateRealDistribution.html">MultivariateRealDistribution</a>
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interface.
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Standard distributions are now available in the
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<a href="https://commons.apache.org/proper/commons-statistics/userguide/index.html">
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Commons Statistics</a> component.
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</p>
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<p>
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An overview of available continuous distributions:<br/>
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<img src="../images/userguide/real_distribution_examples.png" alt="Overview of continuous distributions"/>
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</p>
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</subsection>
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<subsection name="8.2 Distribution Framework" href="distributions">
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<p>
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The distribution framework provides the means to compute probability density
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functions (<code>density(·)</code>), probability mass functions
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(<code>probability(·)</code>) and distribution functions
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(<code>cumulativeProbability(·)</code>) for both
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discrete (integer-valued) and continuous probability distributions.
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The framework also allows for the computation of inverse cumulative probabilities
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and sampling from distributions.
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</p>
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<p>
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For an instance <code>f</code> of a distribution <code>F</code>,
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and a domain value, <code>x</code>, <code>f.cumulativeProbability(x)</code>
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computes <code>P(X <= x)</code> where <code>X</code> is a random variable distributed
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as <code>f</code>, i.e., <code>f.cumulativeProbability(·)</code> represents
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the distribution function of <code>f</code>. If <code>f</code> is continuous,
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(implementing the <code>RealDistribution</code> interface) the probability density
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function of <code>f</code> is represented by <code>f.density(·)</code>.
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For discrete <code>f</code> (implementing <code>IntegerDistribution</code>), the probability
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mass function is represented by <code>f.probability(·)</code>. Continuous
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distributions also implement <code>probability(·)</code> with the same
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definition (<code>f.probability(x)</code> represents <code>P(X = x)</code>
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where <code>X</code> is distributed as <code>f</code>), though in the continuous
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case, this will usually be identically 0.
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</p>
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<source>TDistribution t = new TDistribution(29);
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double lowerTail = t.cumulativeProbability(-2.656); // P(T(29) <= -2.656)
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double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T(29) >= 2.75)</source>
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<p>
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All distributions implement a <code>sample()</code> method to support random sampling from the
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distribution. Implementation classes expose constructors allowing the default
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<a href="../apidocs/org/apache/commons/math4/random/RandomGenerator.html">RandomGenerator</a>
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used by the sampling algorithm to be overridden. If sampling is not going to be used, providing
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a null <code>RandomGenerator</code> constructor argument will avoid the overhead of initializing
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the default generator.
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Commons Math provides
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<ul>
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<li>
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an <a href="../apidocs/org/apache/commons/math4/legacy/distribution/EnumeratedDistribution.html">
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EnumeratedDistribution</a> class that represents discrete distributions of a finite,
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enumerated set of values.
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</li>
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<li>
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a <a href="../apidocs/org/apache/commons/math4/legacy/distribution/MultiVariateNormalDistribution.html">
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MultivariateNormalDistribution</a> interface that represents multivariate Gaussian
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distributions.
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</li>
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</ul>
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</p>
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<p>
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Inverse distribution functions can be computed using the
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<code>inverseCumulativeProbability</code> methods. For continuous <code>f</code>
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and <code>p</code> a probability, <code>f.inverseCumulativeProbability(p)</code> returns
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<code><ul>
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<li>inf{x in R | P(X≤x) ≥ p} for 0 < p < 1},</li>
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<li>inf{x in R | P(X≤x) > 0} for p = 0}.</li>
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<li>inf{x in R | P(X≤x) ≥ p} for 0 < p < 1,</li>
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<li>inf{x in R | P(X≤x) > 0} for p = 0.</li>
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</ul></code> where <code>X</code> is distributed as <code>f</code>.<br/>
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For discrete <code>f</code>, the definition is the same, with <code>Z</code> (the integers)
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in place of <code>R</code>. Note that in the discrete case, the ≥ in the definition
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can make a difference when <code>p</code> is an attained value of the distribution.
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</p>
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</subsection>
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<!--
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TODO: add section on multivariate distributions
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-->
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<subsection name="8.3 User Defined Distributions" href="userdefined">
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<p>
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User-defined distributions can be implemented using
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<a href="../apidocs/org/apache/commons/math4/distribution/RealDistribution.html">RealDistribution</a>,
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<a href="../apidocs/org/apache/commons/math4/distribution/IntegerDistribution.html">IntegerDistribution</a> and
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<a href="../apidocs/org/apache/commons/math4/distribution/MultivariateRealDistribution.html">MultivariateRealDistribution</a>
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interfaces serve as base types. These serve as the basis for all the distributions directly supported by
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Apache Commons Math. To aid in implementing distributions,
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the <a href="../apidocs/org/apache/commons/math4/distribution/AbstractRealDistribution.html">AbstractRealDistribution</a>,
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<a href="../apidocs/org/apache/commons/math4/distribution/AbstractIntegerDistribution.html">AbstractIntegerDistribution</a> and
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<a href="../apidocs/org/apache/commons/math4/distribution/AbstractMultivariateRealDistribution.html">AbstractMultivariateRealDistribution</a>
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provide implementation building blocks and offer basic distribution functionality.
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By extending these abstract classes directly, much of the repetitive distribution
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implementation is already developed and should save time and effort in developing
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user-defined distributions.
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</p>
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</subsection>
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</section>
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</body>
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</document>
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@ -103,8 +103,6 @@
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<li><a href="distribution.html">8. Probability Distributions</a>
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<ul>
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<li><a href="distribution.html#a8.1_Overview">8.1 Overview</a></li>
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<li><a href="distribution.html#a8.2_Distribution_Framework">8.2 Distribution Framework</a></li>
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<li><a href="distribution.html#a8.3_User_Defined_Distributions">8.3 User Defined Distributions</a></li>
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</ul></li>
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<li><a href="fraction.html">9. Fractions</a>
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<ul>
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