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

   1 [section:exp_dist Exponential Distribution]
   2
   3 ``#include <boost/math/distributions/exponential.hpp>``
   4
   5    template <class RealType = double,
   6              class ``__Policy``   = ``__policy_class`` >
   7    class exponential_distribution;
   8
   9    typedef exponential_distribution<> exponential;
  10
  11    template <class RealType, class ``__Policy``>
  12    class exponential_distribution
  13    {
  14    public:
  15       typedef RealType value_type;
  16       typedef Policy   policy_type;
  17
  18       exponential_distribution(RealType lambda = 1);
  19
  20       RealType lambda()const;
  21    };
  22
  23
  24 The [@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution]
  25 is a [@http://en.wikipedia.org/wiki/Probability_distribution continuous probability distribution]
  26 with PDF:
  27
  28 [equation exponential_dist_ref1]
  29
  30 It is often used to model the time between independent
  31 events that happen at a constant average rate.
  32
  33 The following graph shows how the distribution changes for different
  34 values of the rate parameter lambda:
  35
  36 [graph exponential_pdf]
  37
  38 [h4 Member Functions]
  39
  40    exponential_distribution(RealType lambda = 1);
  41
  42 Constructs an
  43 [@http://en.wikipedia.org/wiki/Exponential_distribution Exponential distribution]
  44 with parameter /lambda/.
  45 Lambda is defined as the reciprocal of the scale parameter.
  46
  47 Requires lambda > 0, otherwise calls __domain_error.
  48
  49    RealType lambda()const;
  50
  51 Accessor function returns the lambda parameter of the distribution.
  52
  53 [h4 Non-member Accessors]
  54
  55 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
  56 that are generic to all distributions are supported: __usual_accessors.
  57
  58 The domain of the random variable is \[0, +[infin]\].
  59
  60 [h4 Accuracy]
  61
  62 The exponential distribution is implemented in terms of the
  63 standard library functions `exp`, `log`, `log1p` and `expm1`
  64 and as such should have very low error rates.
  65
  66 [h4 Implementation]
  67
  68 In the following table [lambda] is the parameter lambda of the distribution,
  69 /x/ is the random variate, /p/ is the probability and /q = 1-p/.
  70
  71 [table
  72 [[Function][Implementation Notes]]
  73 [[pdf][Using the relation: pdf = [lambda] * exp(-[lambda] * x) ]]
  74 [[cdf][Using the relation: p = 1 - exp(-x * [lambda]) = -expm1(-x * [lambda]) ]]
  75 [[cdf complement][Using the relation: q = exp(-x * [lambda]) ]]
  76 [[quantile][Using the relation: x = -log(1-p) / [lambda] = -log1p(-p) / [lambda]]]
  77 [[quantile from the complement][Using the relation: x = -log(q) / [lambda]]]
  78 [[mean][1/[lambda]]]
  79 [[standard deviation][1/[lambda]]]
  80 [[mode][0]]
  81 [[skewness][2]]
  82 [[kurtosis][9]]
  83 [[kurtosis excess][6]]
  84 ]
  85
  86 [h4 references]
  87
  88 * [@http://mathworld.wolfram.com/ExponentialDistribution.html Weisstein, Eric W. "Exponential Distribution." From MathWorld--A Wolfram Web Resource]
  89 * [@http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html Wolfram Mathematica calculator]
  90 * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm NIST Exploratory Data Analysis]
  91 * [@http://en.wikipedia.org/wiki/Exponential_distribution Wikipedia Exponential distribution]
  92
  93 (See also the reference documentation for the related __extreme_distrib.)
  94
  95 *
  96 [@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
  97 Samuel Kotz & Saralees Nadarajah]
  98 discuss the relationship of the types of extreme value distributions.
  99
 100 [endsect][/section:exp_dist Exponential]
 101
 102 [/ exponential.qbk
 103   Copyright 2006 John Maddock and Paul A. Bristow.
 104   Distributed under the Boost Software License, Version 1.0.
 105   (See accompanying file LICENSE_1_0.txt or copy at
 106   http://www.boost.org/LICENSE_1_0.txt).
 107 ]
 108