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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.poisson_dist"></a><a class="link" href="poisson_dist.html" title="Poisson Distribution">Poisson Distribution</a>
  28 </h4></div></div></div>
  29 <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>
  30 <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>
  31
  32 <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>
  33           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#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>
  34 <span class="keyword">class</span> <span class="identifier">poisson_distribution</span><span class="special">;</span>
  35
  36 <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>
  37
  38 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">poisson_distribution</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  43   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  44
  45   <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>
  46   <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>
  47 <span class="special">}</span>
  48
  49 <span class="special">}}</span> <span class="comment">// namespaces boost::math</span>
  50 </pre>
  51 <p>
  52           The <a href="http://en.wikipedia.org/wiki/Poisson_distribution" target="_top">Poisson
  53           distribution</a> is a well-known statistical discrete distribution.
  54           It expresses the probability of a number of events (or failures, arrivals,
  55           occurrences ...) occurring in a fixed period of time, provided these events
  56           occur with a known mean rate &#955;
  57 (events/time), and are independent of the
  58           time since the last event.
  59         </p>
  60 <p>
  61           The distribution was discovered by Sim&#233;on-Denis Poisson (1781 to 1840).
  62         </p>
  63 <p>
  64           It has the Probability Mass Function:
  65         </p>
  66 <div class="blockquote"><blockquote class="blockquote"><p>
  67             <span class="inlinemediaobject"><img src="../../../../equations/poisson_ref1.svg"></span>
  68
  69           </p></blockquote></div>
  70 <p>
  71           for k events, with an expected number of events &#955;.
  72         </p>
  73 <p>
  74           The following graph illustrates how the PDF varies with the parameter &#955;:
  75         </p>
  76 <div class="blockquote"><blockquote class="blockquote"><p>
  77             <span class="inlinemediaobject"><img src="../../../../graphs/poisson_pdf_1.svg" align="middle"></span>
  78
  79           </p></blockquote></div>
  80 <div class="caution"><table border="0" summary="Caution">
  81 <tr>
  82 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
  83 <th align="left">Caution</th>
  84 </tr>
  85 <tr><td align="left" valign="top">
  86 <p>
  87             The Poisson distribution is a discrete distribution: internally, functions
  88             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
  89             are continuous functions, but in reality the results returned from these
  90             functions only have meaning if an integer value is provided for the random
  91             variate argument.
  92           </p>
  93 <p>
  94             The quantile function will by default return an integer result that has
  95             been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
  96             (where the probability is less than 0.5) are rounded downward, and upper
  97             quantiles (where the probability is greater than 0.5) are rounded upwards.
  98             This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
  99             least</em></span> the requested coverage will be present in the central
 100             region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
 101             will be present in the tails.
 102           </p>
 103 <p>
 104             This behaviour can be changed so that the quantile functions are rounded
 105             differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
 106             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
 107             Quantiles of Discrete Distributions</a> before using the quantile
 108             function on the Poisson distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
 109             docs</a> describe how to change the rounding policy for these distributions.
 110           </p>
 111 </td></tr>
 112 </table></div>
 113 <h5>
 114 <a name="math_toolkit.dist_ref.dists.poisson_dist.h0"></a>
 115           <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
 116           Functions</a>
 117         </h5>
 118 <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>
 119 </pre>
 120 <p>
 121           Constructs a poisson distribution with mean <span class="emphasis"><em>mean</em></span>.
 122         </p>
 123 <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>
 124 </pre>
 125 <p>
 126           Returns the <span class="emphasis"><em>mean</em></span> of this distribution.
 127         </p>
 128 <h5>
 129 <a name="math_toolkit.dist_ref.dists.poisson_dist.h1"></a>
 130           <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
 131           Accessors</a>
 132         </h5>
 133 <p>
 134           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 135           functions</a> that are generic to all distributions are supported:
 136           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 137           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 138           <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>,
 139           <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>,
 140           <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>,
 141           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 142           <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>,
 143           <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>.
 144         </p>
 145 <p>
 146           The domain of the random variable is [0, &#8734;].
 147         </p>
 148 <h5>
 149 <a name="math_toolkit.dist_ref.dists.poisson_dist.h2"></a>
 150           <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>
 151         </h5>
 152 <p>
 153           The Poisson distribution is implemented in terms of the incomplete gamma
 154           functions <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> and
 155           <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and as such
 156           should have low error rates: but refer to the documentation of those functions
 157           for more information. The quantile and its complement use the inverse gamma
 158           functions and are therefore probably slightly less accurate: this is because
 159           the inverse gamma functions are implemented using an iterative method with
 160           a lower tolerance to avoid excessive computation.
 161         </p>
 162 <h5>
 163 <a name="math_toolkit.dist_ref.dists.poisson_dist.h3"></a>
 164           <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>
 165         </h5>
 166 <p>
 167           In the following table &#955; is the mean of the distribution, <span class="emphasis"><em>k</em></span>
 168           is the random variable, <span class="emphasis"><em>p</em></span> is the probability and
 169           <span class="emphasis"><em>q = 1-p</em></span>.
 170         </p>
 171 <div class="informaltable"><table class="table">
 172 <colgroup>
 173 <col>
 174 <col>
 175 </colgroup>
 176 <thead><tr>
 177 <th>
 178                   <p>
 179                     Function
 180                   </p>
 181                 </th>
 182 <th>
 183                   <p>
 184                     Implementation Notes
 185                   </p>
 186                 </th>
 187 </tr></thead>
 188 <tbody>
 189 <tr>
 190 <td>
 191                   <p>
 192                     pdf
 193                   </p>
 194                 </td>
 195 <td>
 196                   <p>
 197                     Using the relation: pdf = e<sup>-&#955;</sup> &#955;<sup>k</sup> / k!
 198                   </p>
 199                 </td>
 200 </tr>
 201 <tr>
 202 <td>
 203                   <p>
 204                     cdf
 205                   </p>
 206                 </td>
 207 <td>
 208                   <p>
 209                     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,
 210                     &#955;)
 211                   </p>
 212                 </td>
 213 </tr>
 214 <tr>
 215 <td>
 216                   <p>
 217                     cdf complement
 218                   </p>
 219                 </td>
 220 <td>
 221                   <p>
 222                     Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(k+1,
 223                     &#955;)
 224                   </p>
 225                 </td>
 226 </tr>
 227 <tr>
 228 <td>
 229                   <p>
 230                     quantile
 231                   </p>
 232                 </td>
 233 <td>
 234                   <p>
 235                     Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inva</a>(&#955;,
 236                     p) - 1
 237                   </p>
 238                 </td>
 239 </tr>
 240 <tr>
 241 <td>
 242                   <p>
 243                     quantile from the complement
 244                   </p>
 245                 </td>
 246 <td>
 247                   <p>
 248                     Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>(&#955;,
 249                     q) - 1
 250                   </p>
 251                 </td>
 252 </tr>
 253 <tr>
 254 <td>
 255                   <p>
 256                     mean
 257                   </p>
 258                 </td>
 259 <td>
 260                   <p>
 261                     &#955;
 262                   </p>
 263                 </td>
 264 </tr>
 265 <tr>
 266 <td>
 267                   <p>
 268                     mode
 269                   </p>
 270                 </td>
 271 <td>
 272                   <p>
 273                     floor (&#955;) or &#8970;&#955;&#8971;
 274                   </p>
 275                 </td>
 276 </tr>
 277 <tr>
 278 <td>
 279                   <p>
 280                     skewness
 281                   </p>
 282                 </td>
 283 <td>
 284                   <p>
 285                     1/&#8730;&#955;
 286                   </p>
 287                 </td>
 288 </tr>
 289 <tr>
 290 <td>
 291                   <p>
 292                     kurtosis
 293                   </p>
 294                 </td>
 295 <td>
 296                   <p>
 297                     3 + 1/&#955;
 298                   </p>
 299                 </td>
 300 </tr>
 301 <tr>
 302 <td>
 303                   <p>
 304                     kurtosis excess
 305                   </p>
 306                 </td>
 307 <td>
 308                   <p>
 309                     1/&#955;
 310                   </p>
 311                 </td>
 312 </tr>
 313 </tbody>
 314 </table></div>
 315 </div>
 316 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 317 <td align="left"></td>
 318 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 319       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 320       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 321       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 322       Daryle Walker and Xiaogang Zhang<p>
 323         Distributed under the Boost Software License, Version 1.0. (See accompanying
 324         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>)
 325       </p>
 326 </div></td>
 327 </tr></table>
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