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

   1 [section:normal_dist Normal (Gaussian) Distribution]
   2
   3 ``#include <boost/math/distributions/normal.hpp>``
   4
   5    namespace boost{ namespace math{
   6
   7    template <class RealType = double,
   8              class ``__Policy``   = ``__policy_class`` >
   9    class normal_distribution;
  10
  11    typedef normal_distribution<> normal;
  12
  13    template <class RealType, class ``__Policy``>
  14    class normal_distribution
  15    {
  16    public:
  17       typedef RealType value_type;
  18       typedef Policy   policy_type;
  19       // Construct:
  20       normal_distribution(RealType mean = 0, RealType sd = 1);
  21       // Accessors:
  22       RealType mean()const; // location.
  23       RealType standard_deviation()const; // scale.
  24       // Synonyms, provided to allow generic use of find_location and find_scale.
  25       RealType location()const;
  26       RealType scale()const;
  27    };
  28
  29    }} // namespaces
  30
  31 The normal distribution is probably the most well known statistical
  32 distribution: it is also known as the Gaussian Distribution.
  33 A normal distribution with mean zero and standard deviation one
  34 is known as the ['Standard Normal Distribution].
  35
  36 Given mean [mu][space]and standard deviation [sigma][space]it has the PDF:
  37
  38 [space] [equation normal_ref1]
  39
  40 The variation the PDF with its parameters is illustrated
  41 in the following graph:
  42
  43 [graph normal_pdf]
  44
  45 The cumulative distribution function is given by
  46
  47 [space] [equation normal_cdf]
  48
  49 and illustrated by this graph
  50
  51 [graph normal_cdf]
  52
  53
  54 [h4 Member Functions]
  55
  56    normal_distribution(RealType mean = 0, RealType sd = 1);
  57
  58 Constructs a normal distribution with mean /mean/ and
  59 standard deviation /sd/.
  60
  61 Requires sd > 0, otherwise __domain_error is called.
  62
  63    RealType mean()const;
  64    RealType location()const;
  65
  66 both return the /mean/ of this distribution.
  67
  68    RealType standard_deviation()const;
  69    RealType scale()const;
  70
  71 both return the /standard deviation/ of this distribution.
  72 (Redundant location and scale function are provided to match other similar distributions,
  73 allowing the functions find_location and find_scale to be used generically).
  74
  75 [h4 Non-member Accessors]
  76
  77 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
  78 distributions are supported: __usual_accessors.
  79
  80 The domain of the random variable is \[-[max_value], +[min_value]\].
  81 However, the pdf of +[infin] and -[infin] = 0 is also supported,
  82 and cdf at -[infin] = 0, cdf at +[infin] = 1,
  83 and complement cdf -[infin] = 1 and +[infin] = 0,
  84 if RealType permits.
  85
  86 [h4 Accuracy]
  87
  88 The normal distribution is implemented in terms of the
  89 [link math_toolkit.sf_erf.error_function error function],
  90 and as such should have very low error rates.
  91
  92 [h4 Implementation]
  93
  94 In the following table /m/ is the mean of the distribution,
  95 and /s/ is its standard deviation.
  96
  97 [table
  98 [[Function][Implementation Notes]]
  99 [[pdf][Using the relation: pdf = e[super -(x-m)[super 2]\/(2s[super 2])] \/ (s * sqrt(2*pi)) ]]
 100 [[cdf][Using the relation: p = 0.5 * __erfc(-(x-m)/(s*sqrt(2))) ]]
 101 [[cdf complement][Using the relation: q = 0.5 * __erfc((x-m)/(s*sqrt(2))) ]]
 102 [[quantile][Using the relation: x = m - s * sqrt(2) * __erfc_inv(2*p)]]
 103 [[quantile from the complement][Using the relation: x = m + s * sqrt(2) * __erfc_inv(2*p)]]
 104 [[mean and standard deviation][The same as `dist.mean()` and `dist.standard_deviation()`]]
 105 [[mode][The same as the mean.]]
 106 [[median][The same as the mean.]]
 107 [[skewness][0]]
 108 [[kurtosis][3]]
 109 [[kurtosis excess][0]]
 110 ]
 111
 112 [endsect] [/section:normal_dist Normal]
 113
 114 [/ normal.qbk
 115   Copyright 2006, 2007, 2012 John Maddock and Paul A. Bristow.
 116   Distributed under the Boost Software License, Version 1.0.
 117   (See accompanying file LICENSE_1_0.txt or copy at
 118   http://www.boost.org/LICENSE_1_0.txt).
 119 ]
 120