#include <boost/math/distributions/pareto.hpp>

namespace boost{ namespace math{

template <class RealType = double,
          class Policy   = policies::policy<> >
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 Pareto distribution is a continuous distribution with the probability density function (pdf):

f(x; α, β) = αβ^α / x^{α+ 1}

For shape parameter α   > 0, and scale parameter β   > 0. If x < β  , the pdf is zero.

The 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 β:

And this graph illustrates how the PDF varies with the shape parameter α:

Related distributions

Member Functions

pareto_distribution(RealType scale = 1, RealType shape = 1);

Constructs a 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.

Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The supported domain of the random variable is [scale, ∞].

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?.

Implementation

In the following table α   is the shape parameter of the distribution, and β   is its scale parameter, x is the random variate, p is the probability and its complement q = 1-p.

References