...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#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:
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.
All the usual nonmember 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, ∞].
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?.
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 = 1p.
Function 
Implementation Notes 


Using the relation: pdf p = αβ^{α}/x^{α +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 = β / (1  p)^{1/α} 
quantile from the complement 
Using the relation: x = β / (q)^{1/α} 
mean 
αβ / (β  1) 
variance 
βα^{2} / (β  1)^{2} (β  2) 
mode 
α 
skewness 
Refer to Weisstein, Eric W. "Pareto Distribution." From MathWorldA Wolfram Web Resource. 
kurtosis 
Refer to Weisstein, Eric W. "Pareto Distribution." From MathWorldA Wolfram Web Resource. 
kurtosis excess 
Refer to Weisstein, Eric W. "pareto Distribution." From MathWorldA Wolfram Web Resource. 