Fixed some errors; changed to use standard terminology.
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1540566 13f79535-47bb-0310-9956-ffa450edef68
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Phil Steitz
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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 provide a framework for some commonly used
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The distributions package provides a framework and implementations for some commonly used
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probability distributions.
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</p>
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<p>
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@ -38,49 +38,54 @@
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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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function (PDF) probabilities and cumulative distribution function (CDF)
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probabilities for common probability distributions. Along with the direct
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computation of PDF and CDF probabilities, the framework also allows for the
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computation of inverse PDF and inverse CDF values.
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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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</p>
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<p>
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Using a distribution object, PDF and CDF probabilities are easily computed
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using the <code>cumulativeProbability</code> methods. For a distribution
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<code>X</code>, and a domain value, <code>x</code>,
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<code>cumulativeProbability</code> computes <code>P(X <= x)</code>
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(i.e. the lower tail probability of <code>X</code>).
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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 <= -2.656)
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double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)</source>
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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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The inverse PDF and CDF values are just as easily computed using the
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<code>inverseCumulativeProbability</code> methods. For a distribution <code>X</code>,
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and a probability, <code>p</code>, <code>inverseCumulativeProbability</code>
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computes the domain value <code>x</code>, such that:
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<ul>
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<li><code>P(X <= x) = p</code>, for continuous distributions</li>
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<li><code>P(X <= x) <= p</code>, for discrete distributions</li>
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</ul>
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Notice the different cases for continuous and discrete distributions. This is the result
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of PDFs not being invertible functions. As such, for discrete distributions, an exact
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domain value can not be returned. Only the "best" domain value. For Commons-Math, the "best"
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domain value is determined by the largest domain value whose cumulative probability is
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less-than or equal to the given probability.
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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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</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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Since there are numerous distributions and Commons-Math only directly
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supports a handful, it may be necessary to extend the distribution
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framework to satisfy individual needs. It is recommended that the
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User-defined distributions can be implemented using
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<a href="../apidocs/org/apache/commons/math3/distribution/RealDistribution.html">RealDistribution</a>,
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<a href="../apidocs/org/apache/commons/math3/distribution/IntegerDistribution.html">IntegerDistribution</a> and
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<a href="../apidocs/org/apache/commons/math3/distribution/MultivariateRealDistribution.html">MultivariateRealDistribution</a>
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interfaces serve as base types for any extension. These serve as the basis for all the
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distributions directly supported by Commons-Math and using those interfaces
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for implementation purposes will ensure any extension is compatible with the
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remainder of Commons-Math. To aid in implementing a distribution extension,
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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/math3/distribution/AbstractRealDistribution.html">AbstractRealDistribution</a>,
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<a href="../apidocs/org/apache/commons/math3/distribution/AbstractIntegerDistribution.html">AbstractIntegerDistribution</a> and
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<a href="../apidocs/org/apache/commons/math3/distribution/AbstractMultivariateRealDistribution.html">AbstractMultivariateRealDistribution</a>
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