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

   1 [section:uniform_dist Uniform Distribution]
   2
   3
   4 ``#include <boost/math/distributions/uniform.hpp>``
   5
   6    namespace boost{ namespace math{
   7     template <class RealType = double,
   8               class ``__Policy``   = ``__policy_class`` >
   9     class uniform_distribution;
  10
  11     typedef uniform_distribution<> uniform;
  12
  13     template <class RealType, class ``__Policy``>
  14     class uniform_distribution
  15     {
  16     public:
  17        typedef RealType value_type;
  18
  19        uniform_distribution(RealType lower = 0, RealType upper = 1); // Constructor.
  20           : m_lower(lower), m_upper(upper) // Default is standard uniform distribution.
  21        // Accessor functions.
  22        RealType lower()const;
  23        RealType upper()const;
  24     }; // class uniform_distribution
  25
  26    }} // namespaces
  27
  28 The uniform distribution, also known as a rectangular distribution,
  29 is a probability distribution that has constant probability.
  30
  31 The [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 continuous uniform distribution]
  32 is a distribution with the
  33 [@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
  34
  35 f(x) =
  36
  37 * 1 / (upper - lower) for lower < x < upper
  38
  39 * zero for x < lower or x > upper
  40
  41 and in this implementation:
  42
  43 * 1 / (upper - lower) for x = lower or x = upper
  44
  45 The choice of x = lower or x = upper is made because statistical use of this distribution judged is most likely:
  46 the method of maximum likelihood uses this definition.
  47
  48 There is also a [@http://en.wikipedia.org/wiki/Discrete_uniform_distribution *discrete* uniform distribution].
  49
  50 Parameters lower and upper can be any finite value.
  51
  52 The [@http://en.wikipedia.org/wiki/Random_variate random variate]
  53 x must also be finite, and is supported lower <= x <= upper.
  54
  55 The lower parameter is also called the
  56 [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm location parameter],
  57 [@http://en.wikipedia.org/wiki/Location_parameter that is where the origin of a plot will lie],
  58 and (upper - lower) is also called the [@http://en.wikipedia.org/wiki/Scale_parameter scale parameter].
  59
  60 The following graph illustrates how the
  61 [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
  62 varies with the shape parameter:
  63
  64 [graph uniform_pdf]
  65
  66 Likewise for the CDF:
  67
  68 [graph uniform_cdf]
  69
  70 [h4 Member Functions]
  71
  72    uniform_distribution(RealType lower = 0, RealType upper = 1);
  73
  74 Constructs a [@http://en.wikipedia.org/wiki/uniform_distribution
  75 uniform distribution] with lower  /lower/ (a) and upper /upper/ (b).
  76
  77 Requires that the /lower/ and /upper/ parameters are both finite;
  78 otherwise if infinity or NaN then calls __domain_error.
  79
  80    RealType lower()const;
  81
  82 Returns the /lower/ parameter of this distribution.
  83
  84    RealType upper()const;
  85
  86 Returns the /upper/ parameter of this distribution.
  87
  88 [h4 Non-member Accessors]
  89
  90 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
  91 that are generic to all distributions are supported: __usual_accessors.
  92
  93 The domain of the random variable is any finite value,
  94 but the supported range is only /lower/ <= x <= /upper/.
  95
  96 [h4 Accuracy]
  97
  98 The uniform distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two.
  99
 100 [h4 Implementation]
 101
 102 In the following table a is the /lower/ parameter of the distribution,
 103 b is the /upper/ parameter,
 104 /x/ is the random variate, /p/ is the probability and /q = 1-p/.
 105
 106 [table
 107 [[Function][Implementation Notes]]
 108 [[pdf][Using the relation: pdf = 0 for x < a, 1 / (b - a) for a <= x <= b, 0 for x > b ]]
 109 [[cdf][Using the relation: cdf = 0 for x < a, (x - a) / (b - a) for a <= x <= b, 1 for x > b]]
 110 [[cdf complement][Using the relation: q = 1 - p, (b - x) / (b - a) ]]
 111 [[quantile][Using the relation: x = p * (b - a) + a; ]]
 112 [[quantile from the complement][x = -q * (b - a) + b ]]
 113 [[mean][(a + b) / 2 ]]
 114 [[variance][(b - a) [super 2] / 12 ]]
 115 [[mode][any value in \[a, b\] but a is chosen.  (Would NaN be better?) ]]
 116 [[skewness][0]]
 117 [[kurtosis excess][-6/5 = -1.2 exactly. (kurtosis - 3)]]
 118 [[kurtosis][9/5]]
 119 ]
 120
 121 [h4 References]
 122 * [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 Wikpedia continuous uniform distribution]
 123 * [@http://mathworld.wolfram.com/UniformDistribution.html Weisstein, Weisstein, Eric W. "Uniform Distribution." From MathWorld--A Wolfram Web Resource.]
 124 * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm]
 125
 126 [endsect][/section:uniform_dist Uniform]
 127
 128 [/
 129   Copyright 2006 John Maddock and Paul A. Bristow.
 130   Distributed under the Boost Software License, Version 1.0.
 131   (See accompanying file LICENSE_1_0.txt or copy at
 132   http://www.boost.org/LICENSE_1_0.txt).
 133 ]
 134