1 [section:pareto Pareto Distribution]
4 ``#include <boost/math/distributions/pareto.hpp>``
6 namespace boost{ namespace math{
8 template <class RealType = double,
9 class ``__Policy`` = ``__policy_class`` >
10 class pareto_distribution;
12 typedef pareto_distribution<> pareto;
14 template <class RealType, class ``__Policy``>
15 class pareto_distribution
18 typedef RealType value_type;
20 pareto_distribution(RealType scale = 1, RealType shape = 1)
22 RealType scale()const;
23 RealType shape()const;
28 The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution]
29 is a continuous distribution with the
30 [@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]:
32 [expression f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1]]
34 For shape parameter [alpha] > 0, and scale parameter [beta] > 0.
35 If x < [beta], the pdf is zero.
37 The [@http://mathworld.wolfram.com/ParetoDistribution.html Pareto distribution]
38 often describes the larger compared to the smaller.
39 A classic example is that 80% of the wealth is owned by 20% of the population.
41 The following graph illustrates how the PDF varies with the scale parameter [beta]:
45 And this graph illustrates how the PDF varies with the shape parameter [alpha]:
49 [h4 Related distributions]
53 pareto_distribution(RealType scale = 1, RealType shape = 1);
55 Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution
56 pareto distribution] with shape /shape/ and scale /scale/.
58 Requires that the /shape/ and /scale/ parameters are both greater than zero,
59 otherwise calls __domain_error.
61 RealType scale()const;
63 Returns the /scale/ parameter of this distribution.
65 RealType shape()const;
67 Returns the /shape/ parameter of this distribution.
69 [h4 Non-member Accessors]
71 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
72 distributions are supported: __usual_accessors.
74 The supported domain of the random variable is \[scale, [infin]\].
78 The Pareto distribution is implemented in terms of the
79 standard library `exp` functions plus __expm1
80 and so should have very small errors, usually only a few epsilon.
82 If probability is near to unity (or the complement of a probability near zero) see also __why_complements.
86 In the following table [alpha] is the shape parameter of the distribution, and
87 [beta] is its scale parameter, /x/ is the random variate, /p/ is the probability
88 and its complement /q = 1-p/.
91 [[Function][Implementation Notes]]
92 [[pdf][Using the relation: pdf p = [alpha][beta][super [alpha]]/x[super [alpha] +1] ]]
93 [[cdf][Using the relation: cdf p = 1 - ([beta] / x)[super [alpha]] ]]
94 [[cdf complement][Using the relation: q = 1 - p = -([beta] / x)[super [alpha]] ]]
95 [[quantile][Using the relation: x = [beta] / (1 - p)[super 1/[alpha]] ]]
96 [[quantile from the complement][Using the relation: x = [beta] / (q)[super 1/[alpha]] ]]
97 [[mean][[alpha][beta] / ([beta] - 1) ]]
98 [[variance][[beta][alpha][super 2] / ([beta] - 1)[super 2] ([beta] - 2) ]]
100 [[skewness][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
101 [[kurtosis][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
102 [[kurtosis excess][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
106 * [@http://en.wikipedia.org/wiki/pareto_distribution Pareto Distribution]
107 * [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.]
108 * Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267.
109 (Note the meaning of a and b is reversed in Wolfram and Krishnamoorthy).
111 [endsect] [/section:pareto pareto]
114 Copyright 2006, 2009 John Maddock and Paul A. Bristow.
115 Distributed under the Boost Software License, Version 1.0.
116 (See accompanying file LICENSE_1_0.txt or copy at
117 http://www.boost.org/LICENSE_1_0.txt).