ceph/src/boost/libs/math/doc/distributions/pareto.qbk

   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