Weibull Distribution

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##### Weibull Distribution

    #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:

- constant over time, then α = 1, suggests that items are failing from random events.
- decreases over time, then α < 1, suggesting "infant mortality".
- increases over time, then α > 1, suggesting "wear out" - more likely to fail as time goes by.

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:

##### Related distributions

When α = 3, the Weibull distribution appears similar to the normal distribution. When α = 1, the Weibull distribution reduces to the exponential distribution.

##### Member Functions

    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.

##### Non-member Accessors

All the usual non-member 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, ∞].

##### 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.

##### Implementation

In the following table α is the shape parameter of the distribution, β is it's scale parameter, x is the random variate, p is the probability and q = 1-p.

Function | Implementation Notes
--- | ---
pdf | Using the relation: pdf = αβ-α xα - 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 | β2(Γ(1 + 2/α) - Γ2(1 + 1/α))
mode | β((α - 1) / α)1/α
skewness | Refer to Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.
kurtosis | Refer to Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.
kurtosis excess | Refer to Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.

##### References

- http://en.wikipedia.org/wiki/Weibull_distribution
- Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.
- Weibull in NIST Exploratory Data Analysis