...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/distributions/weibull.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > 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 Weibull distribution is a continuous distribution with the probability density function:
f(x; α, β) = (α/β) * (x / β)^{α  1} * e^{(x/β)α}
For shape parameter α > 0, and scale parameter β > 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:
The following graph illustrates how the PDF varies with the shape parameter α:
While this graph illustrates how the PDF varies with the scale parameter β:
When α = 3, the Weibull distribution appears similar to the normal distribution. When α = 1, the Weibull distribution reduces to the exponential distribution. The relationship of the types of extreme value distributions, of which the Weibull is but one, is discussed by Extreme Value Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah.
weibull_distribution(RealType shape, RealType scale = 1);
Constructs a 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.
All the usual nonmember accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
The domain of the random variable is [0, ∞].
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.
In the following table α is the shape parameter of the distribution, β is its scale parameter, x is the random variate, p is the probability and q = 1p.
Function 
Implementation Notes 


Using the relation: pdf = αβ^{α }x^{α  1} e^{(x/beta)alpha} 
cdf 
Using the relation: p = expm1((x/β)^{α}) 
cdf complement 
Using the relation: q = e^{(x/β)α} 
quantile 
Using the relation: x = β * (log1p(p))^{1/α} 
quantile from the complement 
Using the relation: x = β * (log(q))^{1/α} 
mean 
β * Γ(1 + 1/α) 
variance 
β^{2}(Γ(1 + 2/α)  Γ^{2}(1 + 1/α)) 
mode 
β((α  1) / α)^{1/α} 
skewness 
Refer to Weisstein, Eric W. "Weibull Distribution." From MathWorldA Wolfram Web Resource. 
kurtosis 
Refer to Weisstein, Eric W. "Weibull Distribution." From MathWorldA Wolfram Web Resource. 
kurtosis excess 
Refer to Weisstein, Eric W. "Weibull Distribution." From MathWorldA Wolfram Web Resource. 