Mercurial > hg > de.mpg.mpiwg.itgroup.digilib.core
view libs/commons-math-2.1/docs/userguide/distribution.html @ 10:5f2c5fb36e93
commons-math-2.1 added
| author | dwinter |
|---|---|
| date | Tue, 04 Jan 2011 10:00:53 +0100 |
| parents | |
| children |
line wrap: on
line source
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>Math - The Commons Math User Guide - Statistics</title> <style type="text/css" media="all"> @import url("../css/maven-base.css"); @import url("../css/maven-theme.css"); @import url("../css/site.css"); </style> <link rel="stylesheet" href="../css/print.css" type="text/css" media="print" /> <meta http-equiv="Content-Type" content="text/html; charset=ISO-8859-1" /> </head> <body class="composite"> <div id="banner"> <span id="bannerLeft"> Commons Math User Guide </span> <div class="clear"> <hr/> </div> </div> <div id="breadcrumbs"> <div class="xright"> </div> <div class="clear"> <hr/> </div> </div> <div id="leftColumn"> <div id="navcolumn"> <h5>User Guide</h5> <ul> <li class="none"> <a href="../userguide/index.html">Contents</a> </li> <li class="none"> <a href="../userguide/overview.html">Overview</a> </li> <li class="none"> <a href="../userguide/stat.html">Statistics</a> </li> <li class="none"> <a href="../userguide/random.html">Data Generation</a> </li> <li class="none"> <a href="../userguide/linear.html">Linear Algebra</a> </li> <li class="none"> <a href="../userguide/analysis.html">Numerical Analysis</a> </li> <li class="none"> <a href="../userguide/special.html">Special Functions</a> </li> <li class="none"> <a href="../userguide/utilities.html">Utilities</a> </li> <li class="none"> <a href="../userguide/complex.html">Complex Numbers</a> </li> <li class="none"> <strong>Distributions</strong> </li> <li class="none"> <a href="../userguide/fraction.html">Fractions</a> </li> <li class="none"> <a href="../userguide/transform.html">Transform Methods</a> </li> <li class="none"> <a href="../userguide/geometry.html">3D Geometry</a> </li> <li class="none"> <a href="../userguide/optimization.html">Optimization</a> </li> <li class="none"> <a href="../userguide/ode.html">Ordinary Differential Equations</a> </li> <li class="none"> <a href="../userguide/genetics.html">Genetic Algorithms</a> </li> </ul> <a href="http://maven.apache.org/" title="Built by Maven" class="poweredBy"> <img alt="Built by Maven" src="../images/logos/maven-feather.png"></img> </a> </div> </div> <div id="bodyColumn"> <div id="contentBox"> <div class="section"><h2><a name="a8_Probability_Distributions"></a>8 Probability Distributions</h2> <div class="section"><h3><a name="a8.1_Overview"></a>8.1 Overview</h3> <p> The distributions package provide a framework for some commonly used probability distributions. </p> </div> <div class="section"><h3><a name="a8.2_Distribution_Framework"></a>8.2 Distribution Framework</h3> <p> 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. </p> <p> Using a distribution object, PDF and CDF probabilities are easily computed using the <code>cumulativeProbability</code> methods. For a distribution <code>X</code>, and a domain value, <code>x</code>, <code>cumulativeProbability</code> computes <code>P(X <= x)</code> (i.e. the lower tail probability of <code>X</code>). </p> <div class="source"><pre>TDistribution t = new TDistributionImpl(29); double lowerTail = t.cumulativeProbability(-2.656); // P(T <= -2.656) double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)</pre> </div> <p> The inverse PDF and CDF values are just as easily computed using the <code>inverseCumulativeProbability</code> methods. For a distribution <code>X</code>, and a probability, <code>p</code>, <code>inverseCumulativeProbability</code> computes the domain value <code>x</code>, such that: <ul><li><code>P(X <= x) = p</code>, for continuous distributions</li> <li><code>P(X <= x) <= p</code>, for discrete distributions</li> </ul> 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. </p> </div> <div class="section"><h3><a name="a8.3_User_Defined_Distributions"></a>8.3 User Defined Distributions</h3> <p> 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 <code>Distribution</code>, <code>ContinuousDistribution</code>, <code>DiscreteDistribution</code>, and <code>IntegerDistribution</code> 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 insure any extension is compatible with the remainder of Commons-Math. To aid in implementing a distribution extension, the <code>AbstractDistribution</code>, <code>AbstractContinuousDistribution</code>, and <code>AbstractIntegerDistribution</code> provide implementation building blocks and offer a lot of default distribution functionality. By extending these abstract classes directly, a good portion of the repetitive distribution implementation is already developed and should save time and effort in developing user defined distributions. </p> </div> </div> </div> </div> <div class="clear"> <hr/> </div> <div id="footer"> <div class="xright">© 2003-2010 </div> <div class="clear"> <hr/> </div> </div> </body> </html>