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<html>
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<title>Gamma (and Erlang) Distribution</title>
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.gamma_dist"></a><a class="link" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">Gamma (and
Erlang) Distribution</a>
</h4></div></div></div>
<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>
<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="Chapter 14. 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>
<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>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">gamma_distribution</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>
Instead if you want a double precision gamma distribution you can write
</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> <span class="identifier">my_gamma</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span></pre>
</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.svg"></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.svg"></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.svg" align="middle"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/gamma2_pdf.svg" 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>
<h5>
<a name="math_toolkit.dist_ref.dists.gamma_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.gamma_dist.member_functions"></a></span><a class="link" href="gamma_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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>
<h5>
<a name="math_toolkit.dist_ref.dists.gamma_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.gamma_dist.non_member_accessors"></a></span><a class="link" href="gamma_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<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>,
<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>,
<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>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<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>,
<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>.
</p>
<p>
The domain of the random variable is [0,+∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.gamma_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.gamma_dist.accuracy"></a></span><a class="link" href="gamma_dist.html#math_toolkit.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="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> and
<a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their inverses
<a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a> and
<a class="link" href="../../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>
<h5>
<a name="math_toolkit.dist_ref.dists.gamma_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.gamma_dist.implementation"></a></span><a class="link" href="gamma_dist.html#math_toolkit.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 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 <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="../../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="../../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="../../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="../../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="../../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="../../error_handling.html#math_toolkit.error_handling.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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
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