<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Pareto Distribution</title> <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../index.html" title="Math Toolkit"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="normal_dist.html" title="Normal (Gaussian) Distribution"> <link rel="next" href="poisson_dist.html" title="Poisson Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td 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</h5></div></div></div> <p> </p> <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">pareto</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> <p> </p> <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">pareto_distribution</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">pareto_distribution</span><span class="special"><></span> <span class="identifier">pareto</span><span class="special">;</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> <span class="keyword">class</span> <span class="identifier">pareto_distribution</span> <span class="special">{</span> <span class="keyword">public</span><span class="special">:</span> <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> <span class="comment">// Constructor: </span> <span class="identifier">pareto_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">)</span> <span class="comment">// Accessors: </span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="special">};</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> The <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability density function (pdf)</a>: </p> <p> f(x; α, β) = αβ<sup>α</sup> / x<sup>α+ 1</sup> </p> <p> For shape parameter α > 0, and scale parameter β > 0. If x < β, the pdf is zero. </p> <p> The <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Pareto distribution</a> often describes the larger compared to the smaller. A classic example is that 80% of the wealth is owned by 20% of the population. </p> <p> The following graph illustrates how the PDF varies with the scale parameter β: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/pareto_pdf1.png" align="middle"></span> </p> <p> And this graph illustrates how the PDF varies with the shape parameter α: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/pareto_pdf2.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.pareto.related_distributions"></a><h5> <a name="id1057954"></a> <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.related_distributions">Related distributions</a> </h5> <a name="math_toolkit.dist.dist_ref.dists.pareto.member_functions"></a><h5> <a name="id1057972"></a> <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.member_functions">Member Functions</a> </h5> <pre class="programlisting"><span class="identifier">pareto_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> </pre> <p> Constructs a <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">pareto distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>. </p> <p> Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span> parameters are both greater than zero, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> </pre> <p> Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> </pre> <p> Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.pareto.non_member_accessors"></a><h5> <a name="id1058140"></a> <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.non_member_accessors">Non-member Accessors</a> </h5> <p> All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor functions</a> that are generic to all distributions are supported: <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. </p> <p> The supported domain of the random variable is [scale, ∞]. </p> <a name="math_toolkit.dist.dist_ref.dists.pareto.accuracy"></a><h5> <a name="id1058240"></a> <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.accuracy">Accuracy</a> </h5> <p> The Pareto distribution is implemented in terms of the standard library <code class="computeroutput"><span class="identifier">exp</span></code> functions plus <a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a> and so should have very small errors, usually only a few epsilon. </p> <p> If probability is near to unity (or the complement of a probability near zero) see also <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why complements?</a>. </p> <a name="math_toolkit.dist.dist_ref.dists.pareto.implementation"></a><h5> <a name="id1058277"></a> <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.implementation">Implementation</a> </h5> <p> In the following table α is the shape parameter of the distribution, and β is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>. </p> <div class="informaltable"><table class="table"> <colgroup> <col> <col> </colgroup> <thead><tr> <th> <p> Function </p> </th> <th> <p> Implementation Notes </p> </th> </tr></thead> <tbody> <tr> <td> <p> pdf </p> </td> <td> <p> Using the relation: pdf p = αβ<sup>α</sup>/x<sup>α +1</sup> </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relation: cdf p = 1 - (β / x)<sup>α</sup> </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> Using the relation: q = 1 - p = -(β / x)<sup>α</sup> </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> Using the relation: x = β / (1 - p)<sup>1/α</sup> </p> </td> </tr> <tr> <td> <p> quantile from the complement </p> </td> <td> <p> Using the relation: x = β / (q)<sup>1/α</sup> </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> αβ / (β - 1) </p> </td> </tr> <tr> <td> <p> variance </p> </td> <td> <p> βα<sup>2</sup> / (β - 1)<sup>2</sup> (β - 2) </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> α </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.</a> </p> </td> </tr> <tr> <td> <p> kurtosis </p> </td> <td> <p> Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.</a> </p> </td> </tr> <tr> <td> <p> kurtosis excess </p> </td> <td> <p> Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.</a> </p> </td> </tr> </tbody> </table></div> <a name="math_toolkit.dist.dist_ref.dists.pareto.references"></a><h5> <a name="id1058584"></a> <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.references">References</a> </h5> <div class="itemizedlist"><ul type="disc"> <li> <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto Distribution</a> </li> <li> <a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.</a> </li> <li> Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. (Note the meaning of a and b is reversed in Wolfram and Krishnamoorthy). </li> </ul></div> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="normal_dist.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="poisson_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>