<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Gamma (and Erlang) 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="f_dist.html" title="F Distribution"> <link rel="next" href="hypergeometric_dist.html" title="Hypergeometric 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 align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="f_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="hypergeometric_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h5 class="title"> <a name="math_toolkit.dist.dist_ref.dists.gamma_dist"></a><a class="link" href="gamma_dist.html" title="Gamma (and Erlang) Distribution"> Gamma (and Erlang) Distribution</a> </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">gamma</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">gamma_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="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> <span class="identifier">gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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="keyword">const</span><span class="special">;</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="special">};</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> The gamma distribution is a continuous probability distribution. When the shape parameter is an integer then it is known as the Erlang Distribution. It is also closely related to the Poisson and Chi Squared Distributions. </p> <p> When the shape parameter has an integer value, the distribution is the <a href="http://en.wikipedia.org/wiki/Erlang_distribution" target="_top">Erlang distribution</a>. Since this can be produced by ensuring that the shape parameter has an integer value > 0, the Erlang distribution is not separately implemented. </p> <div class="note"><table border="0" summary="Note"> <tr> <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td> <th align="left">Note</th> </tr> <tr><td align="left" valign="top"> <p> To avoid potential confusion with the gamma functions, this distribution does not provide the typedef: </p> <p> </p> <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">gamma_distibution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">gamma</span><span class="special">;</span></pre> <p> </p> <p> Instead if you want a double precision gamma distribution you can use </p> <p> </p> <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gamma_distribution</span><span class="special"><></span></pre> <p> </p> </td></tr> </table></div> <p> For shape parameter <span class="emphasis"><em>k</em></span> and scale parameter θ it is defined by the probability density function: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/gamma_dist_ref1.png"></span> </p> <p> Sometimes an alternative formulation is used: given parameters α= k and β= 1 / θ, then the distribution can be defined by the PDF: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/gamma_dist_ref2.png"></span> </p> <p> In this form the inverse scale parameter is called a <span class="emphasis"><em>rate parameter</em></span>. </p> <p> Both forms are in common usage: this library uses the first definition throughout. Therefore to construct a Gamma Distribution from a <span class="emphasis"><em>rate parameter</em></span>, you should pass the reciprocal of the rate as the scale parameter. </p> <p> The following two graphs illustrate how the PDF of the gamma distribution varies as the parameters vary: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/gamma1_pdf.png" align="middle"></span> </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/gamma2_pdf.png" align="middle"></span> </p> <p> The <span class="bold"><strong>Erlang Distribution</strong></span> is the same as the Gamma, but with the shape parameter an integer. It is often expressed using a <span class="emphasis"><em>rate</em></span> rather than a <span class="emphasis"><em>scale</em></span> as the second parameter (remember that the rate is the reciprocal of the scale). </p> <p> Internally the functions used to implement the Gamma Distribution are already optimised for small-integer arguments, so in general there should be no great loss of performance from using a Gamma Distribution rather than a dedicated Erlang Distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.member_functions"></a><h5> <a name="id1033217"></a> <a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.member_functions">Member Functions</a> </h5> <pre class="programlisting"><span class="identifier">gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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> </pre> <p> Constructs a gamma distribution with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>. </p> <p> Requires that the shape and scale parameters are 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">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> <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> <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.non_member_accessors"></a><h5> <a name="id1033362"></a> <a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.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 domain of the random variable is [0,+∞]. </p> <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.accuracy"></a><h5> <a name="id1035338"></a> <a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.accuracy">Accuracy</a> </h5> <p> The lognormal distribution is implemented in terms of the incomplete gamma functions <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> and <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their inverses <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a> and <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>: refer to the accuracy data for those functions for more information. </p> <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.implementation"></a><h5> <a name="id1035377"></a> <a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.implementation">Implementation</a> </h5> <p> In the following table <span class="emphasis"><em>k</em></span> is the shape parameter of the distribution, θ is it's scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <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 = <a class="link" href="../../../special/sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(k, x / θ) / θ </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relation: p = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(k, x / θ) </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> Using the relation: q = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k, x / θ) </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> Using the relation: x = θ* <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(k, p) </p> </td> </tr> <tr> <td> <p> quantile from the complement </p> </td> <td> <p> Using the relation: x = θ* <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(k, p) </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> kθ </p> </td> </tr> <tr> <td> <p> variance </p> </td> <td> <p> kθ<sup>2</sup> </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> (k-1)θ for <span class="emphasis"><em>k>1</em></span> otherwise a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> 2 / sqrt(k) </p> </td> </tr> <tr> <td> <p> kurtosis </p> </td> <td> <p> 3 + 6 / k </p> </td> </tr> <tr> <td> <p> kurtosis excess </p> </td> <td> <p> 6 / k </p> </td> </tr> </tbody> </table></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="f_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="hypergeometric_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>