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<title>Poisson Distribution</title>
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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.poisson_dist"></a><a class="link" href="poisson_dist.html" title="Poisson Distribution">Poisson 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">poisson</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;14.&#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">poisson_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">poisson_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">poisson</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">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">poisson_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">poisson_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Constructor.</span>
  <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Accessor.</span>
<span class="special">}</span>

<span class="special">}}</span> <span class="comment">// namespaces boost::math</span>
</pre>
<p>
          The <a href="http://en.wikipedia.org/wiki/Poisson_distribution" target="_top">Poisson
          distribution</a> is a well-known statistical discrete distribution.
          It expresses the probability of a number of events (or failures, arrivals,
          occurrences ...) occurring in a fixed period of time, provided these events
          occur with a known mean rate &#955; &#160;
(events/time), and are independent of the
          time since the last event.
        </p>
<p>
          The distribution was discovered by Sim&#233; on-Denis Poisson (1781 to 1840).
        </p>
<p>
          It has the Probability Mass Function:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../equations/poisson_ref1.svg"></span>
        </p>
<p>
          for k events, with an expected number of events &#955;.
        </p>
<p>
          The following graph illustrates how the PDF varies with the parameter &#955;:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../graphs/poisson_pdf_1.svg" align="middle"></span>
        </p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top">
<p>
            The Poisson distribution is a discrete distribution: internally, functions
            like the <code class="computeroutput"><span class="identifier">cdf</span></code> and <code class="computeroutput"><span class="identifier">pdf</span></code> are treated "as if" they
            are continuous functions, but in reality the results returned from these
            functions only have meaning if an integer value is provided for the random
            variate argument.
          </p>
<p>
            The quantile function will by default return an integer result that has
            been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
            (where the probability is less than 0.5) are rounded downward, and upper
            quantiles (where the probability is greater than 0.5) are rounded upwards.
            This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
            least</em></span> the requested coverage will be present in the central
            region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
            will be present in the tails.
          </p>
<p>
            This behaviour can be changed so that the quantile functions are rounded
            differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
            It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
            Quantiles of Discrete Distributions</a> before using the quantile
            function on the Poisson distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
            docs</a> describe how to change the rounding policy for these distributions.
          </p>
</td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.poisson_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.member_functions"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">poisson_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs a poisson distribution with mean <span class="emphasis"><em>mean</em></span>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>mean</em></span> of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.poisson_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.non_member_accessors"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_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.poisson_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.accuracy"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The Poisson 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 as such
          should have low error rates: but refer to the documentation of those functions
          for more information. The quantile and its complement use the inverse gamma
          functions and are therefore probably slightly less accurate: this is because
          the inverse gamma functions are implemented using an iterative method with
          a lower tolerance to avoid excessive computation.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.poisson_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.implementation"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table &#955; &#160; is the mean of the distribution, <span class="emphasis"><em>k</em></span>
          is the random variable, <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 = e<sup>-&#955;</sup> &#955;<sup>k</sup> / k!
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = &#915;(k+1, &#955;) / k! = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k+1,
                    &#955;)
                  </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_p</a>(k+1,
                    &#955;)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inva</a>(&#955;,
                    p) - 1
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>(&#955;,
                    q) - 1
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    &#955;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    floor (&#955;) or &#8970;&#955;&#8971;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    1/&#8730;&#955;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    3 + 1/&#955;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    1/&#955;
                  </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 Nikhar Agrawal,
      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Johan R&#229;de, 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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