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 24The [@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution] 25is a [@http://en.wikipedia.org/wiki/Probability_distribution continuous probability distribution] 26with PDF: 27 28[equation exponential_dist_ref1] 29 30It is often used to model the time between independent 31events that happen at a constant average rate. 32 33The following graph shows how the distribution changes for different 34values of the rate parameter lambda: 35 36[graph exponential_pdf] 37 38[h4 Member Functions] 39 40 exponential_distribution(RealType lambda = 1); 41 42Constructs an 43[@http://en.wikipedia.org/wiki/Exponential_distribution Exponential distribution] 44with parameter /lambda/. 45Lambda is defined as the reciprocal of the scale parameter. 46 47Requires lambda > 0, otherwise calls __domain_error. 48 49 RealType lambda()const; 50 51Accessor function returns the lambda parameter of the distribution. 52 53[h4 Non-member Accessors] 54 55All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] 56that are generic to all distributions are supported: __usual_accessors. 57 58The domain of the random variable is \[0, +[infin]\]. 59 60[h4 Accuracy] 61 62The exponential distribution is implemented in terms of the 63standard library functions `exp`, `log`, `log1p` and `expm1` 64and as such should have very low error rates. 65 66[h4 Implementation] 67 68In 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 97Samuel Kotz & Saralees Nadarajah] 98discuss 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] 108 109