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- [section:weibull_dist Weibull Distribution]
- ``#include <boost/math/distributions/weibull.hpp>``
- namespace boost{ namespace math{
-
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class weibull_distribution;
-
- typedef weibull_distribution<> weibull;
-
- template <class RealType, class ``__Policy``>
- class weibull_distribution
- {
- public:
- typedef RealType value_type;
- typedef Policy policy_type;
- // Construct:
- weibull_distribution(RealType shape, RealType scale = 1)
- // Accessors:
- RealType shape()const;
- RealType scale()const;
- };
-
- }} // namespaces
-
- The [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution]
- is a continuous distribution
- with the
- [@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
- [expression f(x; [alpha], [beta]) = ([alpha]\/[beta]) * (x \/ [beta])[super [alpha] - 1] * e[super -(x\/[beta])[super [alpha]]]]
- For shape parameter ['[alpha]] > 0, and scale parameter ['[beta]] > 0, and /x/ > 0.
- The Weibull distribution is often used in the field of failure analysis;
- in particular it can mimic distributions where the failure rate varies over time.
- If the failure rate is:
- * constant over time, then ['[alpha]] = 1, suggests that items are failing from random events.
- * decreases over time, then ['[alpha]] < 1, suggesting "infant mortality".
- * increases over time, then ['[alpha]] > 1, suggesting "wear out" - more likely to fail as time goes by.
- The following graph illustrates how the PDF varies with the shape parameter ['[alpha]]:
- [graph weibull_pdf1]
- While this graph illustrates how the PDF varies with the scale parameter ['[beta]]:
- [graph weibull_pdf2]
- [h4 Related distributions]
- When ['[alpha]] = 3, the
- [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution] appears similar to the
- [@http://en.wikipedia.org/wiki/Normal_distribution normal distribution].
- When ['[alpha]] = 1, the Weibull distribution reduces to the
- [@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution].
- The relationship of the types of extreme value distributions, of which the Weibull is but one, is
- discussed by
- [@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
- Samuel Kotz & Saralees Nadarajah].
-
- [h4 Member Functions]
- weibull_distribution(RealType shape, RealType scale = 1);
-
- Constructs a [@http://en.wikipedia.org/wiki/Weibull_distribution
- Weibull distribution] with shape /shape/ and scale /scale/.
- Requires that the /shape/ and /scale/ parameters are both greater than zero,
- otherwise calls __domain_error.
- RealType shape()const;
-
- Returns the /shape/ parameter of this distribution.
-
- RealType scale()const;
-
- Returns the /scale/ parameter of this distribution.
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
- distributions are supported: __usual_accessors.
- The domain of the random variable is \[0, [infin]\].
- [h4 Accuracy]
- The Weibull distribution is implemented in terms of the
- standard library `log` and `exp` functions plus __expm1 and __log1p
- and as such should have very low error rates.
- [h4 Implementation]
- In the following table ['[alpha]] is the shape parameter of the distribution,
- ['[beta]] is its scale parameter, /x/ is the random variate, /p/ is the probability
- and /q = 1-p/.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = [alpha][beta][super -[alpha] ]x[super [alpha] - 1] e[super -(x/beta)[super alpha]] ]]
- [[cdf][Using the relation: p = -__expm1(-(x\/[beta])[super [alpha]]) ]]
- [[cdf complement][Using the relation: q = e[super -(x\/[beta])[super [alpha]]] ]]
- [[quantile][Using the relation: x = [beta] * (-__log1p(-p))[super 1\/[alpha]] ]]
- [[quantile from the complement][Using the relation: x = [beta] * (-log(q))[super 1\/[alpha]] ]]
- [[mean][[beta] * [Gamma](1 + 1\/[alpha]) ]]
- [[variance][[beta][super 2]([Gamma](1 + 2\/[alpha]) - [Gamma][super 2](1 + 1\/[alpha])) ]]
- [[mode][[beta](([alpha] - 1) \/ [alpha])[super 1\/[alpha]] ]]
- [[skewness][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
- [[kurtosis][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
- [[kurtosis excess][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
- ]
- [h4 References]
- * [@http://en.wikipedia.org/wiki/Weibull_distribution ]
- * [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.]
- * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm Weibull in NIST Exploratory Data Analysis]
- [endsect] [/section:weibull Weibull]
- [/
- Copyright 2006 John Maddock and Paul A. Bristow.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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