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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.pareto"></a><a class="link" href="pareto.html" title="Pareto Distribution">Pareto Distribution</a>
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</h4></div></div></div>
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<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>
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<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>
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<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>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">pareto_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">pareto_distribution</span><span class="special"><></span> <span class="identifier">pareto</span><span class="special">;</span>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">pareto_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="comment">// Constructor:</span>
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<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>
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<span class="comment">// Accessors:</span>
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<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto
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distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
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density function (pdf)</a>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">f(x; α, β) = αβ<sup>α</sup> / x<sup>α+ 1</sup></span>
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</p></blockquote></div>
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<p>
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For shape parameter α > 0, and scale parameter β > 0. If x < β, the
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pdf is zero.
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</p>
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<p>
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The <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Pareto
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distribution</a> often describes the larger compared to the smaller.
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A classic example is that 80% of the wealth is owned by 20% of the population.
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</p>
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<p>
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The following graph illustrates how the PDF varies with the scale parameter
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β:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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And this graph illustrates how the PDF varies with the shape parameter
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α:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.pareto.h0"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.related_distributions"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.related_distributions">Related
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distributions</a>
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</h5>
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<h5>
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<a name="math_toolkit.dist_ref.dists.pareto.h1"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.member_functions"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.member_functions">Member Functions</a>
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</h5>
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<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>
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</pre>
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<p>
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Constructs a <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">pareto
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distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>.
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</p>
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<p>
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Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span>
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parameters are both greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<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>
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</pre>
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<p>
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Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
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</p>
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<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>
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</pre>
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<p>
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Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.pareto.h2"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.non_member_accessors"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.non_member_accessors">Non-member
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Accessors</a>
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</h5>
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<p>
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All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
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functions</a> that are generic to all distributions are supported:
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
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</p>
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<p>
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The supported domain of the random variable is [scale, ∞].
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.pareto.h3"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.accuracy"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.accuracy">Accuracy</a>
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</h5>
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<p>
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The Pareto distribution is implemented in terms of the standard library
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<code class="computeroutput"><span class="identifier">exp</span></code> functions plus <a class="link" href="../../powers/expm1.html" title="expm1">expm1</a> and so should have very
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small errors, usually only a few epsilon.
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</p>
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<p>
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If probability is near to unity (or the complement of a probability near
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zero) see also <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why complements?</a>.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.pareto.h4"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.implementation"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.implementation">Implementation</a>
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</h5>
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<p>
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In the following table α is the shape parameter of the distribution, and
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β is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate,
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<span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q
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= 1-p</em></span>.
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Implementation Notes
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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pdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: pdf p = αβ<sup>α</sup>/x<sup>α +1</sup>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: cdf p = 1 - (β / x)<sup>α</sup>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf complement
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</p>
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</td>
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<td>
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<p>
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Using the relation: q = 1 - p = -(β / x)<sup>α</sup>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile
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</p>
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</td>
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<td>
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<p>
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Using the relation: x = β / (1 - p)<sup>1/α</sup>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile from the complement
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</p>
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</td>
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<td>
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<p>
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Using the relation: x = β / (q)<sup>1/α</sup>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mean
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</p>
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</td>
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<td>
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<p>
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αβ / (β - 1)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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variance
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</p>
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</td>
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<td>
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<p>
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βα<sup>2</sup> / (β - 1)<sup>2</sup> (β - 2)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mode
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</p>
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</td>
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<td>
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<p>
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α
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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skewness
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</p>
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</td>
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<td>
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<p>
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Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
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Eric W. "Pareto Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis
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</p>
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</td>
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<td>
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<p>
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Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
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Eric W. "Pareto Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis excess
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</p>
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</td>
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<td>
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<p>
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Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
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Eric W. "pareto Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.pareto.h5"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.references"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.references">References</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto
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Distribution</a>
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</li>
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<li class="listitem">
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<a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein,
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Eric W. "Pareto Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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</li>
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<li class="listitem">
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Handbook of Statistical Distributions with Applications, K Krishnamoorthy,
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ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. (Note the meaning of
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a and b is reversed in Wolfram and Krishnamoorthy).
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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