| [section:pareto Pareto Distribution] |
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| ``#include <boost/math/distributions/pareto.hpp>`` |
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| namespace boost{ namespace math{ |
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| template <class RealType = double, |
| class ``__Policy`` = ``__policy_class`` > |
| class pareto_distribution; |
| |
| typedef pareto_distribution<> pareto; |
| |
| template <class RealType, class ``__Policy``> |
| class pareto_distribution |
| { |
| public: |
| typedef RealType value_type; |
| // Constructor: |
| pareto_distribution(RealType scale = 1, RealType shape = 1) |
| // Accessors: |
| RealType scale()const; |
| RealType shape()const; |
| }; |
| |
| }} // namespaces |
| |
| The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution] |
| is a continuous distribution with the |
| [@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]: |
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| f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1] |
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| For shape parameter [alpha][space] > 0, and scale parameter [beta][space] > 0. |
| If x < [beta][space], the pdf is zero. |
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| The [@http://mathworld.wolfram.com/ParetoDistribution.html Pareto distribution] |
| often describes the larger compared to the smaller. |
| A classic example is that 80% of the wealth is owned by 20% of the population. |
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| The following graph illustrates how the PDF varies with the scale parameter [beta]: |
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| [graph pareto_pdf1] |
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| And this graph illustrates how the PDF varies with the shape parameter [alpha]: |
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| [graph pareto_pdf2] |
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| [h4 Related distributions] |
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| [h4 Member Functions] |
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| pareto_distribution(RealType scale = 1, RealType shape = 1); |
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| Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution |
| pareto distribution] with shape /shape/ and scale /scale/. |
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| Requires that the /shape/ and /scale/ parameters are both greater than zero, |
| otherwise calls __domain_error. |
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| RealType scale()const; |
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| Returns the /scale/ parameter of this distribution. |
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| RealType shape()const; |
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| Returns the /shape/ parameter of this distribution. |
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| [h4 Non-member Accessors] |
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| All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all |
| distributions are supported: __usual_accessors. |
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| The supported domain of the random variable is \[scale, [infin]\]. |
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| [h4 Accuracy] |
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| The Pareto distribution is implemented in terms of the |
| standard library `exp` functions plus __expm1 |
| and so should have very small errors, usually only a few epsilon. |
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| If probability is near to unity (or the complement of a probability near zero) see also __why_complements. |
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| [h4 Implementation] |
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| In the following table [alpha][space] is the shape parameter of the distribution, and |
| [beta][space] is its scale parameter, /x/ is the random variate, /p/ is the probability |
| and its complement /q = 1-p/. |
| |
| [table |
| [[Function][Implementation Notes]] |
| [[pdf][Using the relation: pdf p = [alpha][beta][super [alpha]]/x[super [alpha] +1] ]] |
| [[cdf][Using the relation: cdf p = 1 - ([beta][space] / x)[super [alpha]] ]] |
| [[cdf complement][Using the relation: q = 1 - p = -([beta][space] / x)[super [alpha]] ]] |
| [[quantile][Using the relation: x = [beta] / (1 - p)[super 1/[alpha]] ]] |
| [[quantile from the complement][Using the relation: x = [beta] / (q)[super 1/[alpha]] ]] |
| [[mean][[alpha][beta] / ([beta] - 1) ]] |
| [[variance][[beta][alpha][super 2] / ([beta] - 1)[super 2] ([beta] - 2) ]] |
| [[mode][[alpha]]] |
| [[skewness][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] |
| [[kurtosis][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] |
| [[kurtosis excess][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] |
| ] |
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| [h4 References] |
| * [@http://en.wikipedia.org/wiki/pareto_distribution Pareto Distribution] |
| * [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] |
| * Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. |
| (Note the meaning of a and b is reversed in Wolfram and Krishnamoorthy). |
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| [endsect][/section:pareto pareto] |
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| [/ |
| Copyright 2006, 2009 John Maddock and Paul A. Bristow. |
| Distributed under the Boost Software License, Version 1.0. |
| (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt). |
| ] |
| |