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<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.dist.dist_ref.dists.poisson_dist"></a><a class="link" href="poisson_dist.html" title="Poisson Distribution"> Poisson
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">poisson</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">poisson_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">poisson_distribution</span><span class="special"><></span> <span class="identifier">poisson</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">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></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 λ
(events/time), and are independent
of the time since the last event.
</p>
<p>
The distribution was discovered by Simé on-Denis Poisson (1781 to 1840).
</p>
<p>
It has the Probability Mass Function:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/poisson_ref1.png"></span>
</p>
<p>
for k events, with an expected number of events λ.
</p>
<p>
The following graph illustrates how the PDF varies with the parameter
λ:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/poisson_pdf_1.png" align="middle"></span>
</p>
<p>
</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="../../../policy/pol_overview.html" title="Policy Overview">Policies</a>. It is
strongly recommended that you read the tutorial <a class="link" href="../../../policy/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="../../../policy/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>
<p>
</p>
<a name="math_toolkit.dist.dist_ref.dists.poisson_dist.member_functions"></a><h5>
<a name="id1059139"></a>
<a class="link" href="poisson_dist.html#math_toolkit.dist.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>
<a name="math_toolkit.dist.dist_ref.dists.poisson_dist.non_member_accessors"></a><h5>
<a name="id1059228"></a>
<a class="link" href="poisson_dist.html#math_toolkit.dist.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.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.poisson_dist.accuracy"></a><h5>
<a name="id1059325"></a>
<a class="link" href="poisson_dist.html#math_toolkit.dist.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="../../../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 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>
<a name="math_toolkit.dist.dist_ref.dists.poisson_dist.implementation"></a><h5>
<a name="id1059354"></a>
<a class="link" href="poisson_dist.html#math_toolkit.dist.dist_ref.dists.poisson_dist.implementation">Implementation</a>
</h5>
<p>
In the following table λ 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>-λ</sup> λ<sup>k</sup> / k!
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<p>
Using the relation: p = Γ(k+1, λ) / k! = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k+1,
λ)
</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_p</a>(k+1,
λ)
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Using the relation: k = <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inva</a>(λ,
p) - 1
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
Using the relation: k = <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>(λ,
q) - 1
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
λ
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
floor (λ) or ⌊λ⌋
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<p>
1/√λ
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis
</p>
</td>
<td>
<p>
3 + 1/λ
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis excess
</p>
</td>
<td>
<p>
1/λ
</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>
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