pareto.qbk

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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 location = 1, RealType shape = 1)
      // Accessors:
      RealType location()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)]:

f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1]

For shape parameter [alpha][space] > 0, and location parameter [beta][space] > 0, and [alpha][space] > 0.

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 location 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 location = 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 location()const;
      
Returns the /location/ parameter of this distribution.

   RealType shape()const;
   
Returns the /shape/ parameter of this distribution.

[h4 Non-member Accessors]

All the [link math_toolkit.dist.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 \[location, [infin]\].

[h4 Accuracy]

The pareto distribution is implemented in terms of the 
standard library `exp` functions plus __expm1
and as such should have very low error rates
except when probability is very close to unity.

[h4 Implementation]

In the following table [alpha][space] is the shape parameter of the distribution, and
[beta][space] is its location 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 = [alpha] / (1 - p)[super 1/[beta]] ]]
[[quantile from the complement][Using the relation: x =  [alpha] / (q)[super 1/[beta]] ]]
[[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.]

[endsect][/section:pareto pareto]

[/ 
  Copyright 2006 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).
]