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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"><</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> | |
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"><</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 15. 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<></a> <span class="special">></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"><></span> <span class="identifier">poisson</span><span class="special">;</span> | |
37 | ||
38 | <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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 λ   | |
57 | (events/time), and are independent of the | |
58 | time since the last event. | |
59 | </p> | |
60 | <p> | |
61 | The distribution was discovered by Simé on-Denis Poisson (1781 to 1840). | |
62 | </p> | |
63 | <p> | |
64 | It has the Probability Mass Function: | |
65 | </p> | |
66 | <p> | |
67 | <span class="inlinemediaobject"><img src="../../../../equations/poisson_ref1.svg"></span> | |
68 | </p> | |
69 | <p> | |
70 | for k events, with an expected number of events λ. | |
71 | </p> | |
72 | <p> | |
73 | The following graph illustrates how the PDF varies with the parameter λ: | |
74 | </p> | |
75 | <p> | |
76 | <span class="inlinemediaobject"><img src="../../../../graphs/poisson_pdf_1.svg" align="middle"></span> | |
77 | </p> | |
78 | <div class="caution"><table border="0" summary="Caution"> | |
79 | <tr> | |
80 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> | |
81 | <th align="left">Caution</th> | |
82 | </tr> | |
83 | <tr><td align="left" valign="top"> | |
84 | <p> | |
85 | The Poisson distribution is a discrete distribution: internally, functions | |
86 | 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 | |
87 | are continuous functions, but in reality the results returned from these | |
88 | functions only have meaning if an integer value is provided for the random | |
89 | variate argument. | |
90 | </p> | |
91 | <p> | |
92 | The quantile function will by default return an integer result that has | |
93 | been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles | |
94 | (where the probability is less than 0.5) are rounded downward, and upper | |
95 | quantiles (where the probability is greater than 0.5) are rounded upwards. | |
96 | This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at | |
97 | least</em></span> the requested coverage will be present in the central | |
98 | region, and <span class="emphasis"><em>no more than</em></span> the requested coverage | |
99 | will be present in the tails. | |
100 | </p> | |
101 | <p> | |
102 | This behaviour can be changed so that the quantile functions are rounded | |
103 | differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>. | |
104 | 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 | |
105 | Quantiles of Discrete Distributions</a> before using the quantile | |
106 | function on the Poisson distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference | |
107 | docs</a> describe how to change the rounding policy for these distributions. | |
108 | </p> | |
109 | </td></tr> | |
110 | </table></div> | |
111 | <h5> | |
112 | <a name="math_toolkit.dist_ref.dists.poisson_dist.h0"></a> | |
113 | <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 | |
114 | Functions</a> | |
115 | </h5> | |
116 | <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> | |
117 | </pre> | |
118 | <p> | |
119 | Constructs a poisson distribution with mean <span class="emphasis"><em>mean</em></span>. | |
120 | </p> | |
121 | <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> | |
122 | </pre> | |
123 | <p> | |
124 | Returns the <span class="emphasis"><em>mean</em></span> of this distribution. | |
125 | </p> | |
126 | <h5> | |
127 | <a name="math_toolkit.dist_ref.dists.poisson_dist.h1"></a> | |
128 | <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 | |
129 | Accessors</a> | |
130 | </h5> | |
131 | <p> | |
132 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
133 | functions</a> that are generic to all distributions are supported: | |
134 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
135 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
136 | <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>, | |
137 | <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>, | |
138 | <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>, | |
139 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
140 | <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>, | |
141 | <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>. | |
142 | </p> | |
143 | <p> | |
144 | The domain of the random variable is [0, ∞]. | |
145 | </p> | |
146 | <h5> | |
147 | <a name="math_toolkit.dist_ref.dists.poisson_dist.h2"></a> | |
148 | <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> | |
149 | </h5> | |
150 | <p> | |
151 | The Poisson distribution is implemented in terms of the incomplete gamma | |
152 | functions <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> and | |
153 | <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and as such | |
154 | should have low error rates: but refer to the documentation of those functions | |
155 | for more information. The quantile and its complement use the inverse gamma | |
156 | functions and are therefore probably slightly less accurate: this is because | |
157 | the inverse gamma functions are implemented using an iterative method with | |
158 | a lower tolerance to avoid excessive computation. | |
159 | </p> | |
160 | <h5> | |
161 | <a name="math_toolkit.dist_ref.dists.poisson_dist.h3"></a> | |
162 | <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> | |
163 | </h5> | |
164 | <p> | |
165 | In the following table λ   is the mean of the distribution, <span class="emphasis"><em>k</em></span> | |
166 | is the random variable, <span class="emphasis"><em>p</em></span> is the probability and | |
167 | <span class="emphasis"><em>q = 1-p</em></span>. | |
168 | </p> | |
169 | <div class="informaltable"><table class="table"> | |
170 | <colgroup> | |
171 | <col> | |
172 | <col> | |
173 | </colgroup> | |
174 | <thead><tr> | |
175 | <th> | |
176 | <p> | |
177 | Function | |
178 | </p> | |
179 | </th> | |
180 | <th> | |
181 | <p> | |
182 | Implementation Notes | |
183 | </p> | |
184 | </th> | |
185 | </tr></thead> | |
186 | <tbody> | |
187 | <tr> | |
188 | <td> | |
189 | <p> | |
190 | ||
191 | </p> | |
192 | </td> | |
193 | <td> | |
194 | <p> | |
195 | Using the relation: pdf = e<sup>-λ</sup> λ<sup>k</sup> / k! | |
196 | </p> | |
197 | </td> | |
198 | </tr> | |
199 | <tr> | |
200 | <td> | |
201 | <p> | |
202 | cdf | |
203 | </p> | |
204 | </td> | |
205 | <td> | |
206 | <p> | |
207 | Using the relation: p = Γ(k+1, λ) / k! = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k+1, | |
208 | λ) | |
209 | </p> | |
210 | </td> | |
211 | </tr> | |
212 | <tr> | |
213 | <td> | |
214 | <p> | |
215 | cdf complement | |
216 | </p> | |
217 | </td> | |
218 | <td> | |
219 | <p> | |
220 | Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(k+1, | |
221 | λ) | |
222 | </p> | |
223 | </td> | |
224 | </tr> | |
225 | <tr> | |
226 | <td> | |
227 | <p> | |
228 | quantile | |
229 | </p> | |
230 | </td> | |
231 | <td> | |
232 | <p> | |
233 | Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inva</a>(λ, | |
234 | p) - 1 | |
235 | </p> | |
236 | </td> | |
237 | </tr> | |
238 | <tr> | |
239 | <td> | |
240 | <p> | |
241 | quantile from the complement | |
242 | </p> | |
243 | </td> | |
244 | <td> | |
245 | <p> | |
246 | Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>(λ, | |
247 | q) - 1 | |
248 | </p> | |
249 | </td> | |
250 | </tr> | |
251 | <tr> | |
252 | <td> | |
253 | <p> | |
254 | mean | |
255 | </p> | |
256 | </td> | |
257 | <td> | |
258 | <p> | |
259 | λ | |
260 | </p> | |
261 | </td> | |
262 | </tr> | |
263 | <tr> | |
264 | <td> | |
265 | <p> | |
266 | mode | |
267 | </p> | |
268 | </td> | |
269 | <td> | |
270 | <p> | |
271 | floor (λ) or ⌊λ⌋ | |
272 | </p> | |
273 | </td> | |
274 | </tr> | |
275 | <tr> | |
276 | <td> | |
277 | <p> | |
278 | skewness | |
279 | </p> | |
280 | </td> | |
281 | <td> | |
282 | <p> | |
283 | 1/√λ | |
284 | </p> | |
285 | </td> | |
286 | </tr> | |
287 | <tr> | |
288 | <td> | |
289 | <p> | |
290 | kurtosis | |
291 | </p> | |
292 | </td> | |
293 | <td> | |
294 | <p> | |
295 | 3 + 1/λ | |
296 | </p> | |
297 | </td> | |
298 | </tr> | |
299 | <tr> | |
300 | <td> | |
301 | <p> | |
302 | kurtosis excess | |
303 | </p> | |
304 | </td> | |
305 | <td> | |
306 | <p> | |
307 | 1/λ | |
308 | </p> | |
309 | </td> | |
310 | </tr> | |
311 | </tbody> | |
312 | </table></div> | |
313 | </div> | |
314 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
315 | <td align="left"></td> | |
316 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
317 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
318 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
319 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
320 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
321 | 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>) | |
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