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  25 <div class="section">
  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>
 343 </div></td>
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