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<html>
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<title>Gamma (and Erlang) Distribution</title>
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<div class="section math_toolkit_dist_dist_ref_dists_gamma_dist">
<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">&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>
<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">&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="Policies">Policy</a>   <span class="special">=</span> <a class="link" href="../../../policy/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>
<p>
</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>
            </p>
<p>
              Instead if you want a double precision gamma distribution you can write
            </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">&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>
<p>
            </p>
</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.png"></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.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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h0"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.member_functions"></a></span><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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h1"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.non_member_accessors"></a></span><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,+&#8734;].
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h2"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.accuracy"></a></span><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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h3"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.implementation"></a></span><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, &#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="../../../special/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="../../../special/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="../../../special/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="../../../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 = &#952; &#160;* <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>
                      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="../../../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>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
      Lalande, Johan R&#229;de, 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>
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