Weibull Distribution |
(continuous probability dist for equations) |
Usage: |
WeibullDist (x, α, β) |
Definition: |
(α/β) (x/β)^(α-1) exp (-(x/β)^α) |
Parameters: |
α is shape β is scale |
Required: |
α > 0 β > 0 |
Support: |
x ⋝ 0 |
Moments: |
μ = β gamma (1 + 1/α) σ^2 = β^2 [gamma (1 + 2/α) - gamma (1 + 1/α) ^ 2] g1 = [gamma (1+3/α) - 3 gamma (1+1/α) gamma (1+2/α) + 2 gamma (1+1/α)^3] / [gamma (1+2/α) – gamma (1+1/α)^2]^(3/2) β2 = [gamma (1+4/α) - 4 gamma (1+1/α) gamma (1+3/α) + 6 gamma (1+1/α)^2 gamma (1+2/α) - 3 gamma (1+1/α)^4] / [gamma (1+2/α) - gamma (1+1/α)^2]^2 |
The Weibull distribution is often used for reliability models, since if the failure rate of an item (i.e., percent of the remaining ones which fail, as a function of time) is given as: Ζ(t) = r t^(α-1), then the distribution of item lifetimes is given by the Weibull distribution with r = α / β ^ α.