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<title>Gamma (and Erlang) Distribution</title>
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<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">&lt;</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">&gt;</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">&lt;</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&#160;18.&#160;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&lt;&gt;</a> <span class="special">&gt;</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 &gt; 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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;&gt;</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 &#952; &#160; 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 &#945; &#160;= k and
&#946; &#160;= 1 / &#952;, 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,+&#8734;].
</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 gamma 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, &#952; &#160; 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 / &#952;) / &#952;
</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 / &#952;)
</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 / &#952;)
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Using the relation: x = &#952; &#160;* <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 = &#952; &#160;* <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>
k&#952;
</p>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<p>
k&#952;<sup>2</sup>
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
(k-1)&#952; &#160; for <span class="emphasis"><em>k&gt;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 &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
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