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<document url= "stat.html" >
<properties >
<title > The Commons Math User Guide - Statistics</title>
</properties>
<body >
<section name= "8 Probability Distributions" >
<subsection name= "8.1 Overview" href= "overview" >
<p >
The distributions package provide a framework for some commonly used
probability distributions.
</p>
</subsection>
<subsection name= "8.2 Distribution Framework" href= "distributions" >
<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 >
In order to use the distribution framework, first a distribution object must
be created. It is encouraged that all distribution object creation occurs via
the <code > org.apache.commons.math.distribution.DistributionFactory</code>
class. <code > DistributionFactory</code> is a simple factory used to create all
of the distribution objects supported by Commons-Math. The typical usage of
<code > DistributionFactory</code> to create a distribution object would be:
</p>
<source > DistributionFactory factory = DistributionFactory.newInstance();
BinomialDistribution binomial = factory.createBinomialDistribution(10, .75);</source>
<p >
The distributions that can be instantiated via the <code > DistributionFactory</code>
are detailed below:
<table >
<tr > <th > Distribution</th> <th > Factory Method</th> <th > Parameters</th> </tr>
<tr > <td > Binomial</td> <td > createBinomialDistribution</td> <td > <div > Number of trials</div> <div > Probability of success</div> </td> </tr>
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<tr > <td > Cauchy</td> <td > createCauchyDistribution</td> <td > <div > Median</div> <div > Scale</div> </td> </tr>
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<tr > <td > Chi-Squared</td> <td > createChiSquaredDistribution</td> <td > <div > Degrees of freedom</div> </td> </tr>
<tr > <td > Exponential</td> <td > createExponentialDistribution</td> <td > <div > Mean</div> </td> </tr>
<tr > <td > F</td> <td > createFDistribution</td> <td > <div > Numerator degrees of freedom</div> <div > Denominator degrees of freedom</div> </td> </tr>
<tr > <td > Gamma</td> <td > createGammaDistribution</td> <td > <div > Alpha</div> <div > Beta</div> </td> </tr>
<tr > <td > Hypergeometric</td> <td > createHypogeometricDistribution</td> <td > <div > Population size</div> <div > Number of successes in population</div> <div > Sample size</div> </td> </tr>
<tr > <td > Normal (Gaussian)</td> <td > createNormalDistribution</td> <td > <div > Mean</div> <div > Standard Deviation</div> </td> </tr>
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<tr > <td > Poisson</td> <td > createPoissonDistribution</td> <td > <div > Mean</div> </td> </tr>
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<tr > <td > t</td> <td > createTDistribution</td> <td > <div > Degrees of freedom</div> </td> </tr>
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<tr > <td > Weibull</td> <td > createWeibullDistribution</td> <td > <div > Shape</div> <div > Scale</div> <div > Location</div> </td> </tr>
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</table>
</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>
<source > DistributionFactory factory = DistributionFactory.newInstance();
TDistribution t = factory.createBinomialDistribution(29);
double lowerTail = t.cumulativeProbability(-2.656); // P(T < = -2.656)
double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T > = 2.75)</source>
<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>
</subsection>
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<subsection name= "8.3 User Defined Distributions" href= "userdefined" >
<p >
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Since there are numerous distributions and Commons-Math only directly supports a handful,
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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
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repetitive distribution implementation is already developed and should save time and effort
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in developing user defined distributions.
</p>
</subsection>
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</section>
</body>
</document>
