1[section:weibull_dist Weibull Distribution] 2 3``#include <boost/math/distributions/weibull.hpp>`` 4 5 namespace boost{ namespace math{ 6 7 template <class RealType = double, 8 class ``__Policy`` = ``__policy_class`` > 9 class weibull_distribution; 10 11 typedef weibull_distribution<> weibull; 12 13 template <class RealType, class ``__Policy``> 14 class weibull_distribution 15 { 16 public: 17 typedef RealType value_type; 18 typedef Policy policy_type; 19 // Construct: 20 weibull_distribution(RealType shape, RealType scale = 1) 21 // Accessors: 22 RealType shape()const; 23 RealType scale()const; 24 }; 25 26 }} // namespaces 27 28The [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution] 29is a continuous distribution 30with the 31[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]: 32 33[expression f(x; [alpha], [beta]) = ([alpha]\/[beta]) * (x \/ [beta])[super [alpha] - 1] * e[super -(x\/[beta])[super [alpha]]]] 34 35For shape parameter ['[alpha]] > 0, and scale parameter ['[beta]] > 0, and /x/ > 0. 36 37The Weibull distribution is often used in the field of failure analysis; 38in particular it can mimic distributions where the failure rate varies over time. 39If the failure rate is: 40 41* constant over time, then ['[alpha]] = 1, suggests that items are failing from random events. 42* decreases over time, then ['[alpha]] < 1, suggesting "infant mortality". 43* increases over time, then ['[alpha]] > 1, suggesting "wear out" - more likely to fail as time goes by. 44 45The following graph illustrates how the PDF varies with the shape parameter ['[alpha]]: 46 47[graph weibull_pdf1] 48 49While this graph illustrates how the PDF varies with the scale parameter ['[beta]]: 50 51[graph weibull_pdf2] 52 53[h4 Related distributions] 54 55When ['[alpha]] = 3, the 56[@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution] appears similar to the 57[@http://en.wikipedia.org/wiki/Normal_distribution normal distribution]. 58When ['[alpha]] = 1, the Weibull distribution reduces to the 59[@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution]. 60The relationship of the types of extreme value distributions, of which the Weibull is but one, is 61discussed by 62[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications 63Samuel Kotz & Saralees Nadarajah]. 64 65 66[h4 Member Functions] 67 68 weibull_distribution(RealType shape, RealType scale = 1); 69 70Constructs a [@http://en.wikipedia.org/wiki/Weibull_distribution 71Weibull distribution] with shape /shape/ and scale /scale/. 72 73Requires that the /shape/ and /scale/ parameters are both greater than zero, 74otherwise calls __domain_error. 75 76 RealType shape()const; 77 78Returns the /shape/ parameter of this distribution. 79 80 RealType scale()const; 81 82Returns the /scale/ parameter of this distribution. 83 84[h4 Non-member Accessors] 85 86All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all 87distributions are supported: __usual_accessors. 88 89The domain of the random variable is \[0, [infin]\]. 90 91[h4 Accuracy] 92 93The Weibull distribution is implemented in terms of the 94standard library `log` and `exp` functions plus __expm1 and __log1p 95and as such should have very low error rates. 96 97[h4 Implementation] 98 99 100In the following table ['[alpha]] is the shape parameter of the distribution, 101['[beta]] is its scale parameter, /x/ is the random variate, /p/ is the probability 102and /q = 1-p/. 103 104[table 105[[Function][Implementation Notes]] 106[[pdf][Using the relation: pdf = [alpha][beta][super -[alpha] ]x[super [alpha] - 1] e[super -(x/beta)[super alpha]] ]] 107[[cdf][Using the relation: p = -__expm1(-(x\/[beta])[super [alpha]]) ]] 108[[cdf complement][Using the relation: q = e[super -(x\/[beta])[super [alpha]]] ]] 109[[quantile][Using the relation: x = [beta] * (-__log1p(-p))[super 1\/[alpha]] ]] 110[[quantile from the complement][Using the relation: x = [beta] * (-log(q))[super 1\/[alpha]] ]] 111[[mean][[beta] * [Gamma](1 + 1\/[alpha]) ]] 112[[variance][[beta][super 2]([Gamma](1 + 2\/[alpha]) - [Gamma][super 2](1 + 1\/[alpha])) ]] 113[[mode][[beta](([alpha] - 1) \/ [alpha])[super 1\/[alpha]] ]] 114[[skewness][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] 115[[kurtosis][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] 116[[kurtosis excess][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] 117] 118 119[h4 References] 120* [@http://en.wikipedia.org/wiki/Weibull_distribution ] 121* [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] 122* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm Weibull in NIST Exploratory Data Analysis] 123 124[endsect] [/section:weibull Weibull] 125 126[/ 127 Copyright 2006 John Maddock and Paul A. Bristow. 128 Distributed under the Boost Software License, Version 1.0. 129 (See accompanying file LICENSE_1_0.txt or copy at 130 http://www.boost.org/LICENSE_1_0.txt). 131] 132