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- [section:pareto Pareto Distribution]
- ``#include <boost/math/distributions/pareto.hpp>``
- namespace boost{ namespace math{
- 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)]:
- [expression f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1]]
- For shape parameter [alpha] > 0, and scale parameter [beta] > 0.
- If x < [beta], the pdf is zero.
- 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.
- The following graph illustrates how the PDF varies with the scale parameter [beta]:
- [graph pareto_pdf1]
- And this graph illustrates how the PDF varies with the shape parameter [alpha]:
- [graph pareto_pdf2]
- [h4 Related distributions]
- [h4 Member Functions]
- pareto_distribution(RealType scale = 1, RealType shape = 1);
- Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution
- pareto distribution] with shape /shape/ and scale /scale/.
- Requires that the /shape/ and /scale/ parameters are both greater than zero,
- otherwise calls __domain_error.
- RealType scale()const;
- Returns the /scale/ parameter of this distribution.
- RealType shape()const;
- Returns the /shape/ parameter of this distribution.
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
- distributions are supported: __usual_accessors.
- The supported domain of the random variable is \[scale, [infin]\].
- [h4 Accuracy]
- 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.
- If probability is near to unity (or the complement of a probability near zero) see also __why_complements.
- [h4 Implementation]
- In the following table [alpha] is the shape parameter of the distribution, and
- [beta] 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] / x)[super [alpha]] ]]
- [[cdf complement][Using the relation: q = 1 - p = -([beta] / 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.] ]]
- ]
- [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).
- [endsect] [/section:pareto pareto]
- [/
- 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).
- ]
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