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Posted to commits@commons.apache.org by ps...@apache.org on 2013/11/10 23:19:10 UTC
svn commit: r1540566 -
/commons/proper/math/trunk/src/site/xdoc/userguide/distribution.xml
Author: psteitz
Date: Sun Nov 10 22:19:10 2013
New Revision: 1540566
URL: http://svn.apache.org/r1540566
Log:
Fixed some errors; changed to use standard terminology.
Modified:
commons/proper/math/trunk/src/site/xdoc/userguide/distribution.xml
Modified: commons/proper/math/trunk/src/site/xdoc/userguide/distribution.xml
URL: http://svn.apache.org/viewvc/commons/proper/math/trunk/src/site/xdoc/userguide/distribution.xml?rev=1540566&r1=1540565&r2=1540566&view=diff
==============================================================================
--- commons/proper/math/trunk/src/site/xdoc/userguide/distribution.xml (original)
+++ commons/proper/math/trunk/src/site/xdoc/userguide/distribution.xml Sun Nov 10 22:19:10 2013
@@ -27,7 +27,7 @@
<section name="8 Probability Distributions">
<subsection name="8.1 Overview" href="overview">
<p>
- The distributions package provide a framework for some commonly used
+ The distributions package provides a framework and implementations for some commonly used
probability distributions.
</p>
<p>
@@ -38,49 +38,54 @@
<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>
- 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>).
+ functions (<code>density(·)</code>), probability mass functions
+ (<code>probability(·)</code>) and distribution functions
+ (<code>cumulativeProbability(·)</code>) for both
+ discrete (integer-valued) and continuous probability distributions.
+ The framework also allows for the computation of inverse cumulative probabilities.
+ </p>
+ <p>
+ For an instance <code>f</code> of a distribution <code>F</code>,
+ and a domain value, <code>x</code>, <code>f.cumulativeProbability(x)</code>
+ computes <code>P(X <= x)</code> where <code>X</code> is a random variable distributed
+ as <code>f</code>, i.e., <code>f.cumulativeProbability(·)</code> represents
+ the distribution function of <code>f</code>. If <code>f</code> is continuous,
+ (implementing the <code>RealDistribution</code> interface) the probability density
+ function of <code>f</code> is represented by <code>f.density(·)</code>.
+ For discrete <code>f</code> (implementing <code>IntegerDistribution</code>), the probability
+ mass function is represented by <code>f.probability(·)</code>. Continuous
+ distributions also implement <code>probability(·)</code> with the same
+ definition (<code>f.probability(x)</code> represents <code>P(X = x)</code>
+ where <code>X</code> is distributed as <code>f</code>), though in the continuous
+ case, this will usually be identically 0.
</p>
<source>TDistribution t = new TDistribution(29);
-double lowerTail = t.cumulativeProbability(-2.656); // P(T <= -2.656)
-double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)</source>
+double lowerTail = t.cumulativeProbability(-2.656); // P(T(29) <= -2.656)
+double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T(29) >= 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.
+ Inverse distribution functions can be computed using the
+ <code>inverseCumulativeProbability</code> methods. For continuous <code>f</code>
+ and <code>p</code> a probability, <code>f.inverseCumulativeProbability(p)</code> returns
+ <code><ul>
+ <li>inf{x in R | P(X≤x) ≥ p} for 0 < p < 1},</li>
+ <li>inf{x in R | P(X≤x) > 0} for p = 0}.</li>
+ </ul></code> where <code>X</code> is distributed as <code>f</code>.<br/>
+ For discrete <code>f</code>, the definition is the same, with <code>Z</code> (the integers)
+ in place of <code>R</code>. Note that in the discrete case, the ≥ in the definition
+ can make a difference when <code>p</code> is an attained value of the distribution.
</p>
</subsection>
+ <!--
+ TODO: add section on multivariate distributions
+ -->
<subsection name="8.3 User Defined Distributions" href="userdefined">
<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
+ User-defined distributions can be implemented using
<a href="../apidocs/org/apache/commons/math3/distribution/RealDistribution.html">RealDistribution</a>,
<a href="../apidocs/org/apache/commons/math3/distribution/IntegerDistribution.html">IntegerDistribution</a> and
<a href="../apidocs/org/apache/commons/math3/distribution/MultivariateRealDistribution.html">MultivariateRealDistribution</a>
- 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 ensure any extension is compatible with the
- remainder of Commons-Math. To aid in implementing a distribution extension,
+ interfaces serve as base types. These serve as the basis for all the distributions directly supported by
+ Apache Commons Math. To aid in implementing distributions,
the <a href="../apidocs/org/apache/commons/math3/distribution/AbstractRealDistribution.html">AbstractRealDistribution</a>,
<a href="../apidocs/org/apache/commons/math3/distribution/AbstractIntegerDistribution.html">AbstractIntegerDistribution</a> and
<a href="../apidocs/org/apache/commons/math3/distribution/AbstractMultivariateRealDistribution.html">AbstractMultivariateRealDistribution</a>