Pareto Distribution

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##### Pareto Distribution

    #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 location = 1, RealType shape = 1)
       // Accessors:
       RealType location()const;
       RealType shape()const;
    };

    }} // namespaces

The Pareto distribution is a continuous distribution with the probability density function (pdf):

f(x; α, β) = αβα / xα+ 1

For shape parameter α > 0, and location parameter β > 0, and α > 0.

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 shape parameter α:

Pareto pdf

##### Related distributions

##### Member Functions

    pareto_distribution(RealType location = 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 location()const;

Returns the location 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 [location, ∞].

##### 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.

##### Implementation

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

Function | Implementation Notes
--- | ---
pdf | Using the relation: pdf p = αβα/xα +1
cdf | Using the relation: cdf p = 1 - (β / x)α
cdf complement | Using the relation: q = 1 - p = -(β / x)α
quantile | Using the relation: x = α / (1 - p)1/β
quantile from the complement | Using the relation: x = α / (q)1/β
mean | αβ / (β - 1)
variance | βα2 / (β - 1)2 (β - 2)
mode | α
skewness | Refer to Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.
kurtosis | Refer to Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.
kurtosis excess | Refer to Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.

##### References

- Pareto Distribution
- Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.