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<document url="distribution.html">
  <properties>
    <title>The Commons Math User Guide - Distributions</title>
  </properties>
  <body>
    <section name="8 Probability Distributions">
      <subsection name="8.1 Overview" href="overview">
        <p>
          Standard distributions are now available in the
          <a href="https://commons.apache.org/proper/commons-statistics/userguide/index.html">
          Commons Statistics</a> component.
        </p>
        <p>
          Commons Math provides
          <ul>
            <li>
              an <a href="../apidocs/org/apache/commons/math4/legacy/distribution/EnumeratedDistribution.html">
              EnumeratedDistribution</a> class that represents discrete distributions of a finite,
              enumerated set of values.
            </li>
            <li>
              a <a href="../apidocs/org/apache/commons/math4/legacy/distribution/MultiVariateNormalDistribution.html">
              MultivariateNormalDistribution</a> interface that represents multivariate Gaussian
              distributions.
            </li>
          </ul>
        </p>
        <p>
          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&le;x) &ge; p} for 0 &lt; p &lt; 1,</li>
            <li>inf{x in R | P(X&le;x) &gt; 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 &ge; in the definition
          can make a difference when <code>p</code> is an attained value of the distribution.
        </p>
      </subsection>

      <subsection name="8.2 Generating data like an input file"
                  href="empirical">
        <p>
          Using the <code>EmpiricalDistribution</code> class, you can generate data based on
          the values in an input file:

          <source>
int binCount = 500;
EmpiricalDistribution empDist = new EmpiricalDistribution(binCount);
empDist.load("data.txt");
RealDistribution.Sampler sampler = empDist.createSampler(RandomSource.MT.create());
double value = sampler.nextDouble(); </source>

          The entire input file is read and a probability density function is estimated
          based on data from the file.
          The estimation method is essentially the
          <a href="http://nedwww.ipac.caltech.edu/level5/March02/Silverman/Silver2_6.html">
            Variable Kernel Method</a> with Gaussian smoothing.
          The created sampler will return random values whose probability distribution
          matches the empirical distribution (i.e. if you generate a large number of
          such values, their distribution should "look like" the distribution of the
          values in the input file.
          The values are not stored in memory in this case either, so there is no limit to the
          size of the input file.
        </p>
      </subsection>
    </section>
  </body>
</document>