Mercurial > hg > de.mpg.mpiwg.itgroup.digilib.core
diff libs/commons-math-2.1/docs/userguide/distribution.html @ 10:5f2c5fb36e93
commons-math-2.1 added
| author | dwinter |
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| date | Tue, 04 Jan 2011 10:00:53 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/libs/commons-math-2.1/docs/userguide/distribution.html Tue Jan 04 10:00:53 2011 +0100 @@ -0,0 +1,232 @@ +<!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>