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26 <div class=
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"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"><</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">></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>
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">pareto_distribution
</span><span class=
"special">;
</span>
36 <span class=
"keyword">typedef
</span> <span class=
"identifier">pareto_distribution
</span><span class=
"special"><></span> <span class=
"identifier">pareto
</span><span class=
"special">;
</span>
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">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>
50 <span class=
"special">}}
</span> <span class=
"comment">// namespaces
</span>
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>:
58 f(x;
α,
β) =
αβ<sup>α</sup> / x
<sup>α+
1</sup>
61 For shape parameter
α   > 0, and scale parameter
β   > 0. If x
< β  , the
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.
70 The following graph illustrates how the PDF varies with the scale parameter
74 <span class=
"inlinemediaobject"><img src=
"../../../../graphs/pareto_pdf1.svg" align=
"middle"></span>
77 And this graph illustrates how the PDF varies with the shape parameter
81 <span class=
"inlinemediaobject"><img src=
"../../../../graphs/pareto_pdf2.svg" align=
"middle"></span>
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
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>
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>
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>.
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>.
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>
105 Returns the
<span class=
"emphasis"><em>scale
</em></span> parameter of this distribution.
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>
110 Returns the
<span class=
"emphasis"><em>shape
</em></span> parameter of this distribution.
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
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>.
130 The supported domain of the random variable is [scale,
∞].
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>
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.
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>.
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>
150 In the following table
α   is the shape parameter of the distribution, and
151 β   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
155 <div class=
"informaltable"><table class=
"table">
181 Using the relation: pdf p =
αβ<sup>α</sup>/x
<sup>α +
1</sup>
193 Using the relation: cdf p =
1 - (
β   / x)
<sup>α</sup>
205 Using the relation: q =
1 - p = -(
β   / x)
<sup>α</sup>
217 Using the relation: x =
β / (
1 - p)
<sup>1/
α</sup>
224 quantile from the complement
229 Using the relation: x =
β / (q)
<sup>1/
α</sup>
241 αβ / (
β -
1)
253 βα<sup>2</sup> / (
β -
1)
<sup>2</sup> (
β -
2)
277 Refer to
<a href=
"http://mathworld.wolfram.com/ParetoDistribution.html" target=
"_top">Weisstein,
278 Eric W.
"Pareto Distribution." From MathWorld--A Wolfram
291 Refer to
<a href=
"http://mathworld.wolfram.com/ParetoDistribution.html" target=
"_top">Weisstein,
292 Eric W.
"Pareto Distribution." From MathWorld--A Wolfram
305 Refer to
<a href=
"http://mathworld.wolfram.com/ParetoDistribution.html" target=
"_top">Weisstein,
306 Eric W.
"pareto Distribution." From MathWorld--A Wolfram
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>
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
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
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).
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
© 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
åde, 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>)
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