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26<div class="titlepage"><div><div><h4 class="title">
27<a name="math_toolkit.dist_ref.dists.pareto"></a><a class="link" href="pareto.html" title="Pareto Distribution">Pareto 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">pareto</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;15.&#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">pareto_distribution</span><span class="special">;</span>
35
36<span class="keyword">typedef</span> <span class="identifier">pareto_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">pareto</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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
39<span class="keyword">class</span> <span class="identifier">pareto_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="comment">// Constructor:</span>
44 <span class="identifier">pareto_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">)</span>
45 <span class="comment">// Accessors:</span>
46 <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
47 <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
48<span class="special">};</span>
49
50<span class="special">}}</span> <span class="comment">// namespaces</span>
51</pre>
52<p>
53 The <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto
54 distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
55 density function (pdf)</a>:
56 </p>
57<p>
58 f(x; &#945;, &#946;) = &#945;&#946;<sup>&#945;</sup> / x<sup>&#945;+ 1</sup>
59 </p>
60<p>
61 For shape parameter &#945; &#160; &gt; 0, and scale parameter &#946; &#160; &gt; 0. If x &lt; &#946; &#160;, the
62 pdf is zero.
63 </p>
64<p>
65 The <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Pareto
66 distribution</a> often describes the larger compared to the smaller.
67 A classic example is that 80% of the wealth is owned by 20% of the population.
68 </p>
69<p>
70 The following graph illustrates how the PDF varies with the scale parameter
71 &#946;:
72 </p>
73<p>
74 <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.svg" align="middle"></span>
75 </p>
76<p>
77 And this graph illustrates how the PDF varies with the shape parameter
78 &#945;:
79 </p>
80<p>
81 <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.svg" align="middle"></span>
82 </p>
83<h5>
84<a name="math_toolkit.dist_ref.dists.pareto.h0"></a>
85 <span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.related_distributions"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.related_distributions">Related
86 distributions</a>
87 </h5>
88<h5>
89<a name="math_toolkit.dist_ref.dists.pareto.h1"></a>
90 <span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.member_functions"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.member_functions">Member Functions</a>
91 </h5>
92<pre class="programlisting"><span class="identifier">pareto_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
93</pre>
94<p>
95 Constructs a <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">pareto
96 distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>.
97 </p>
98<p>
99 Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span>
100 parameters are both greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
101 </p>
102<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
103</pre>
104<p>
105 Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
106 </p>
107<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
108</pre>
109<p>
110 Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution.
111 </p>
112<h5>
113<a name="math_toolkit.dist_ref.dists.pareto.h2"></a>
114 <span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.non_member_accessors"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.non_member_accessors">Non-member
115 Accessors</a>
116 </h5>
117<p>
118 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
119 functions</a> that are generic to all distributions are supported:
120 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
121 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
122 <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>,
123 <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>,
124 <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>,
125 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
126 <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>,
127 <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>.
128 </p>
129<p>
130 The supported domain of the random variable is [scale, &#8734;].
131 </p>
132<h5>
133<a name="math_toolkit.dist_ref.dists.pareto.h3"></a>
134 <span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.accuracy"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.accuracy">Accuracy</a>
135 </h5>
136<p>
137 The Pareto distribution is implemented in terms of the standard library
138 <code class="computeroutput"><span class="identifier">exp</span></code> functions plus <a class="link" href="../../powers/expm1.html" title="expm1">expm1</a> and so should have very
139 small errors, usually only a few epsilon.
140 </p>
141<p>
142 If probability is near to unity (or the complement of a probability near
143 zero) see also <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why complements?</a>.
144 </p>
145<h5>
146<a name="math_toolkit.dist_ref.dists.pareto.h4"></a>
147 <span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.implementation"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.implementation">Implementation</a>
148 </h5>
149<p>
150 In the following table &#945; &#160; is the shape parameter of the distribution, and
151 &#946; &#160; is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate,
152 <span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q
153 = 1-p</em></span>.
154 </p>
155<div class="informaltable"><table class="table">
156<colgroup>
157<col>
158<col>
159</colgroup>
160<thead><tr>
161<th>
162 <p>
163 Function
164 </p>
165 </th>
166<th>
167 <p>
168 Implementation Notes
169 </p>
170 </th>
171</tr></thead>
172<tbody>
173<tr>
174<td>
175 <p>
176 pdf
177 </p>
178 </td>
179<td>
180 <p>
181 Using the relation: pdf p = &#945;&#946;<sup>&#945;</sup>/x<sup>&#945; +1</sup>
182 </p>
183 </td>
184</tr>
185<tr>
186<td>
187 <p>
188 cdf
189 </p>
190 </td>
191<td>
192 <p>
193 Using the relation: cdf p = 1 - (&#946; &#160; / x)<sup>&#945;</sup>
194 </p>
195 </td>
196</tr>
197<tr>
198<td>
199 <p>
200 cdf complement
201 </p>
202 </td>
203<td>
204 <p>
205 Using the relation: q = 1 - p = -(&#946; &#160; / x)<sup>&#945;</sup>
206 </p>
207 </td>
208</tr>
209<tr>
210<td>
211 <p>
212 quantile
213 </p>
214 </td>
215<td>
216 <p>
217 Using the relation: x = &#946; / (1 - p)<sup>1/&#945;</sup>
218 </p>
219 </td>
220</tr>
221<tr>
222<td>
223 <p>
224 quantile from the complement
225 </p>
226 </td>
227<td>
228 <p>
229 Using the relation: x = &#946; / (q)<sup>1/&#945;</sup>
230 </p>
231 </td>
232</tr>
233<tr>
234<td>
235 <p>
236 mean
237 </p>
238 </td>
239<td>
240 <p>
241 &#945;&#946; / (&#946; - 1)
242 </p>
243 </td>
244</tr>
245<tr>
246<td>
247 <p>
248 variance
249 </p>
250 </td>
251<td>
252 <p>
253 &#946;&#945;<sup>2</sup> / (&#946; - 1)<sup>2</sup> (&#946; - 2)
254 </p>
255 </td>
256</tr>
257<tr>
258<td>
259 <p>
260 mode
261 </p>
262 </td>
263<td>
264 <p>
265 &#945;
266 </p>
267 </td>
268</tr>
269<tr>
270<td>
271 <p>
272 skewness
273 </p>
274 </td>
275<td>
276 <p>
277 Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
278 Eric W. "Pareto Distribution." From MathWorld--A Wolfram
279 Web Resource.</a>
280 </p>
281 </td>
282</tr>
283<tr>
284<td>
285 <p>
286 kurtosis
287 </p>
288 </td>
289<td>
290 <p>
291 Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
292 Eric W. "Pareto Distribution." From MathWorld--A Wolfram
293 Web Resource.</a>
294 </p>
295 </td>
296</tr>
297<tr>
298<td>
299 <p>
300 kurtosis excess
301 </p>
302 </td>
303<td>
304 <p>
305 Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
306 Eric W. "pareto Distribution." From MathWorld--A Wolfram
307 Web Resource.</a>
308 </p>
309 </td>
310</tr>
311</tbody>
312</table></div>
313<h5>
314<a name="math_toolkit.dist_ref.dists.pareto.h5"></a>
315 <span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.references"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.references">References</a>
316 </h5>
317<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
318<li class="listitem">
319 <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto
320 Distribution</a>
321 </li>
322<li class="listitem">
323 <a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein,
324 Eric W. "Pareto Distribution." From MathWorld--A Wolfram
325 Web Resource.</a>
326 </li>
327<li class="listitem">
328 Handbook of Statistical Distributions with Applications, K Krishnamoorthy,
329 ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. (Note the meaning of
330 a and b is reversed in Wolfram and Krishnamoorthy).
331 </li>
332</ul></div>
333</div>
334<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
335<td align="left"></td>
336<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
337 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
338 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
339 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
340 Distributed under the Boost Software License, Version 1.0. (See accompanying
341 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>)
342 </p>
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