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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;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">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;20.&#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 <div class="blockquote"><blockquote class="blockquote"><p>
  58             <span class="serif_italic">f(x; &#945;, &#946;) = &#945;&#946;<sup>&#945;</sup> / x<sup>&#945;+ 1</sup></span>
  59           </p></blockquote></div>
  60 <p>
  61           For shape parameter &#945; &gt; 0, and scale parameter &#946; &gt; 0. If x &lt; &#946;, 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 <div class="blockquote"><blockquote class="blockquote"><p>
  74             <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.svg" align="middle"></span>
  75
  76           </p></blockquote></div>
  77 <p>
  78           And this graph illustrates how the PDF varies with the shape parameter
  79           &#945;:
  80         </p>
  81 <div class="blockquote"><blockquote class="blockquote"><p>
  82             <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.svg" align="middle"></span>
  83
  84           </p></blockquote></div>
  85 <h5>
  86 <a name="math_toolkit.dist_ref.dists.pareto.h0"></a>
  87           <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
  88           distributions</a>
  89         </h5>
  90 <h5>
  91 <a name="math_toolkit.dist_ref.dists.pareto.h1"></a>
  92           <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>
  93         </h5>
  94 <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>
  95 </pre>
  96 <p>
  97           Constructs a <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">pareto
  98           distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>.
  99         </p>
 100 <p>
 101           Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span>
 102           parameters are both greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 103         </p>
 104 <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>
 105 </pre>
 106 <p>
 107           Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
 108         </p>
 109 <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>
 110 </pre>
 111 <p>
 112           Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution.
 113         </p>
 114 <h5>
 115 <a name="math_toolkit.dist_ref.dists.pareto.h2"></a>
 116           <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
 117           Accessors</a>
 118         </h5>
 119 <p>
 120           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 121           functions</a> that are generic to all distributions are supported:
 122           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 123           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 124           <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>,
 125           <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>,
 126           <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>,
 127           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 128           <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>,
 129           <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>.
 130         </p>
 131 <p>
 132           The supported domain of the random variable is [scale, &#8734;].
 133         </p>
 134 <h5>
 135 <a name="math_toolkit.dist_ref.dists.pareto.h3"></a>
 136           <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>
 137         </h5>
 138 <p>
 139           The Pareto distribution is implemented in terms of the standard library
 140           <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
 141           small errors, usually only a few epsilon.
 142         </p>
 143 <p>
 144           If probability is near to unity (or the complement of a probability near
 145           zero) see also <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why complements?</a>.
 146         </p>
 147 <h5>
 148 <a name="math_toolkit.dist_ref.dists.pareto.h4"></a>
 149           <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>
 150         </h5>
 151 <p>
 152           In the following table &#945; is the shape parameter of the distribution, and
 153           &#946; is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate,
 154           <span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q
 155           = 1-p</em></span>.
 156         </p>
 157 <div class="informaltable"><table class="table">
 158 <colgroup>
 159 <col>
 160 <col>
 161 </colgroup>
 162 <thead><tr>
 163 <th>
 164                   <p>
 165                     Function
 166                   </p>
 167                 </th>
 168 <th>
 169                   <p>
 170                     Implementation Notes
 171                   </p>
 172                 </th>
 173 </tr></thead>
 174 <tbody>
 175 <tr>
 176 <td>
 177                   <p>
 178                     pdf
 179                   </p>
 180                 </td>
 181 <td>
 182                   <p>
 183                     Using the relation: pdf p = &#945;&#946;<sup>&#945;</sup>/x<sup>&#945; +1</sup>
 184                   </p>
 185                 </td>
 186 </tr>
 187 <tr>
 188 <td>
 189                   <p>
 190                     cdf
 191                   </p>
 192                 </td>
 193 <td>
 194                   <p>
 195                     Using the relation: cdf p = 1 - (&#946; / x)<sup>&#945;</sup>
 196                   </p>
 197                 </td>
 198 </tr>
 199 <tr>
 200 <td>
 201                   <p>
 202                     cdf complement
 203                   </p>
 204                 </td>
 205 <td>
 206                   <p>
 207                     Using the relation: q = 1 - p = -(&#946; / x)<sup>&#945;</sup>
 208                   </p>
 209                 </td>
 210 </tr>
 211 <tr>
 212 <td>
 213                   <p>
 214                     quantile
 215                   </p>
 216                 </td>
 217 <td>
 218                   <p>
 219                     Using the relation: x = &#946; / (1 - p)<sup>1/&#945;</sup>
 220                   </p>
 221                 </td>
 222 </tr>
 223 <tr>
 224 <td>
 225                   <p>
 226                     quantile from the complement
 227                   </p>
 228                 </td>
 229 <td>
 230                   <p>
 231                     Using the relation: x = &#946; / (q)<sup>1/&#945;</sup>
 232                   </p>
 233                 </td>
 234 </tr>
 235 <tr>
 236 <td>
 237                   <p>
 238                     mean
 239                   </p>
 240                 </td>
 241 <td>
 242                   <p>
 243                     &#945;&#946; / (&#946; - 1)
 244                   </p>
 245                 </td>
 246 </tr>
 247 <tr>
 248 <td>
 249                   <p>
 250                     variance
 251                   </p>
 252                 </td>
 253 <td>
 254                   <p>
 255                     &#946;&#945;<sup>2</sup> / (&#946; - 1)<sup>2</sup> (&#946; - 2)
 256                   </p>
 257                 </td>
 258 </tr>
 259 <tr>
 260 <td>
 261                   <p>
 262                     mode
 263                   </p>
 264                 </td>
 265 <td>
 266                   <p>
 267                     &#945;
 268                   </p>
 269                 </td>
 270 </tr>
 271 <tr>
 272 <td>
 273                   <p>
 274                     skewness
 275                   </p>
 276                 </td>
 277 <td>
 278                   <p>
 279                     Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
 280                     Eric W. "Pareto Distribution." From MathWorld--A Wolfram
 281                     Web Resource.</a>
 282                   </p>
 283                 </td>
 284 </tr>
 285 <tr>
 286 <td>
 287                   <p>
 288                     kurtosis
 289                   </p>
 290                 </td>
 291 <td>
 292                   <p>
 293                     Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
 294                     Eric W. "Pareto Distribution." From MathWorld--A Wolfram
 295                     Web Resource.</a>
 296                   </p>
 297                 </td>
 298 </tr>
 299 <tr>
 300 <td>
 301                   <p>
 302                     kurtosis excess
 303                   </p>
 304                 </td>
 305 <td>
 306                   <p>
 307                     Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein,
 308                     Eric W. "pareto Distribution." From MathWorld--A Wolfram
 309                     Web Resource.</a>
 310                   </p>
 311                 </td>
 312 </tr>
 313 </tbody>
 314 </table></div>
 315 <h5>
 316 <a name="math_toolkit.dist_ref.dists.pareto.h5"></a>
 317           <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>
 318         </h5>
 319 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 320 <li class="listitem">
 321               <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto
 322               Distribution</a>
 323             </li>
 324 <li class="listitem">
 325               <a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein,
 326               Eric W. "Pareto Distribution." From MathWorld--A Wolfram
 327               Web Resource.</a>
 328             </li>
 329 <li class="listitem">
 330               Handbook of Statistical Distributions with Applications, K Krishnamoorthy,
 331               ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. (Note the meaning of
 332               a and b is reversed in Wolfram and Krishnamoorthy).
 333             </li>
 334 </ul></div>
 335 </div>
 336 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 337 <td align="left"></td>
 338 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 339       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 340       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 341       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 342       Daryle Walker and Xiaogang Zhang<p>
 343         Distributed under the Boost Software License, Version 1.0. (See accompanying
 344         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>)
 345       </p>
 346 </div></td>
 347 </tr></table>
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