1[section:pareto Pareto Distribution] 2 3 4``#include <boost/math/distributions/pareto.hpp>`` 5 6 namespace boost{ namespace math{ 7 8 template <class RealType = double, 9 class ``__Policy`` = ``__policy_class`` > 10 class pareto_distribution; 11 12 typedef pareto_distribution<> pareto; 13 14 template <class RealType, class ``__Policy``> 15 class pareto_distribution 16 { 17 public: 18 typedef RealType value_type; 19 // Constructor: 20 pareto_distribution(RealType scale = 1, RealType shape = 1) 21 // Accessors: 22 RealType scale()const; 23 RealType shape()const; 24 }; 25 26 }} // namespaces 27 28The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution] 29is a continuous distribution with the 30[@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]: 31 32[expression f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1]] 33 34For shape parameter [alpha] > 0, and scale parameter [beta] > 0. 35If x < [beta], the pdf is zero. 36 37The [@http://mathworld.wolfram.com/ParetoDistribution.html Pareto distribution] 38often describes the larger compared to the smaller. 39A classic example is that 80% of the wealth is owned by 20% of the population. 40 41The following graph illustrates how the PDF varies with the scale parameter [beta]: 42 43[graph pareto_pdf1] 44 45And this graph illustrates how the PDF varies with the shape parameter [alpha]: 46 47[graph pareto_pdf2] 48 49[h4 Related distributions] 50 51[h4 Member Functions] 52 53 pareto_distribution(RealType scale = 1, RealType shape = 1); 54 55Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution 56pareto distribution] with shape /shape/ and scale /scale/. 57 58Requires that the /shape/ and /scale/ parameters are both greater than zero, 59otherwise calls __domain_error. 60 61 RealType scale()const; 62 63Returns the /scale/ parameter of this distribution. 64 65 RealType shape()const; 66 67Returns the /shape/ parameter of this distribution. 68 69[h4 Non-member Accessors] 70 71All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all 72distributions are supported: __usual_accessors. 73 74The supported domain of the random variable is \[scale, [infin]\]. 75 76[h4 Accuracy] 77 78The Pareto distribution is implemented in terms of the 79standard library `exp` functions plus __expm1 80and so should have very small errors, usually only a few epsilon. 81 82If probability is near to unity (or the complement of a probability near zero) see also __why_complements. 83 84[h4 Implementation] 85 86In 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 88and its complement /q = 1-p/. 89 90[table 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) ]] 99[[mode][[alpha]]] 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.] ]] 103] 104 105[h4 References] 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). 110 111[endsect] [/section:pareto pareto] 112 113[/ 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). 118] 119 120