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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="id1054709"></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="id1054798"></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="id1054895"></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="id1054924"></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, 2010 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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