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1 | [section:pareto Pareto Distribution] |
2 | ||
3 | ||
4 | ``#include <boost/math/distributions/pareto.hpp>`` | |
5 | ||
6 | namespace boost{ namespace math{ | |
7 | ||
8 | template <class RealType = double, | |
9 | class ``__Policy`` = ``__policy_class`` > | |
10 | class pareto_distribution; | |
11 | ||
12 | typedef pareto_distribution<> pareto; | |
13 | ||
14 | template <class RealType, class ``__Policy``> | |
15 | class pareto_distribution | |
16 | { | |
17 | public: | |
18 | typedef RealType value_type; | |
19 | // Constructor: | |
20 | pareto_distribution(RealType scale = 1, RealType shape = 1) | |
21 | // Accessors: | |
22 | RealType scale()const; | |
23 | RealType shape()const; | |
24 | }; | |
25 | ||
26 | }} // namespaces | |
27 | ||
28 | The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution] | |
29 | is a continuous distribution with the | |
30 | [@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]: | |
31 | ||
32 | f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1] | |
33 | ||
34 | For shape parameter [alpha][space] > 0, and scale parameter [beta][space] > 0. | |
35 | If x < [beta][space], the pdf is zero. | |
36 | ||
37 | The [@http://mathworld.wolfram.com/ParetoDistribution.html Pareto distribution] | |
38 | often describes the larger compared to the smaller. | |
39 | A classic example is that 80% of the wealth is owned by 20% of the population. | |
40 | ||
41 | The following graph illustrates how the PDF varies with the scale parameter [beta]: | |
42 | ||
43 | [graph pareto_pdf1] | |
44 | ||
45 | And this graph illustrates how the PDF varies with the shape parameter [alpha]: | |
46 | ||
47 | [graph pareto_pdf2] | |
48 | ||
49 | ||
50 | [h4 Related distributions] | |
51 | ||
52 | ||
53 | [h4 Member Functions] | |
54 | ||
55 | pareto_distribution(RealType scale = 1, RealType shape = 1); | |
56 | ||
57 | Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution | |
58 | pareto distribution] with shape /shape/ and scale /scale/. | |
59 | ||
60 | Requires that the /shape/ and /scale/ parameters are both greater than zero, | |
61 | otherwise calls __domain_error. | |
62 | ||
63 | RealType scale()const; | |
64 | ||
65 | Returns the /scale/ parameter of this distribution. | |
66 | ||
67 | RealType shape()const; | |
68 | ||
69 | Returns the /shape/ parameter of this distribution. | |
70 | ||
71 | [h4 Non-member Accessors] | |
72 | ||
73 | All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all | |
74 | distributions are supported: __usual_accessors. | |
75 | ||
76 | The supported domain of the random variable is \[scale, [infin]\]. | |
77 | ||
78 | [h4 Accuracy] | |
79 | ||
80 | The Pareto distribution is implemented in terms of the | |
81 | standard library `exp` functions plus __expm1 | |
82 | and so should have very small errors, usually only a few epsilon. | |
83 | ||
84 | If probability is near to unity (or the complement of a probability near zero) see also __why_complements. | |
85 | ||
86 | [h4 Implementation] | |
87 | ||
88 | In the following table [alpha][space] is the shape parameter of the distribution, and | |
89 | [beta][space] is its scale parameter, /x/ is the random variate, /p/ is the probability | |
90 | and its complement /q = 1-p/. | |
91 | ||
92 | [table | |
93 | [[Function][Implementation Notes]] | |
94 | [[pdf][Using the relation: pdf p = [alpha][beta][super [alpha]]/x[super [alpha] +1] ]] | |
95 | [[cdf][Using the relation: cdf p = 1 - ([beta][space] / x)[super [alpha]] ]] | |
96 | [[cdf complement][Using the relation: q = 1 - p = -([beta][space] / x)[super [alpha]] ]] | |
97 | [[quantile][Using the relation: x = [beta] / (1 - p)[super 1/[alpha]] ]] | |
98 | [[quantile from the complement][Using the relation: x = [beta] / (q)[super 1/[alpha]] ]] | |
99 | [[mean][[alpha][beta] / ([beta] - 1) ]] | |
100 | [[variance][[beta][alpha][super 2] / ([beta] - 1)[super 2] ([beta] - 2) ]] | |
101 | [[mode][[alpha]]] | |
102 | [[skewness][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] | |
103 | [[kurtosis][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] | |
104 | [[kurtosis excess][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] | |
105 | ] | |
106 | ||
107 | [h4 References] | |
108 | * [@http://en.wikipedia.org/wiki/pareto_distribution Pareto Distribution] | |
109 | * [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] | |
110 | * Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. | |
111 | (Note the meaning of a and b is reversed in Wolfram and Krishnamoorthy). | |
112 | ||
113 | [endsect][/section:pareto pareto] | |
114 | ||
115 | [/ | |
116 | Copyright 2006, 2009 John Maddock and Paul A. Bristow. | |
117 | Distributed under the Boost Software License, Version 1.0. | |
118 | (See accompanying file LICENSE_1_0.txt or copy at | |
119 | http://www.boost.org/LICENSE_1_0.txt). | |
120 | ] | |
121 |