Weibulldistribution

The Weibull distribution, sometimes spelled Weibull distribution, is a continuous probability distribution used to model non-negative real-valued lifetimes. In its common two-parameter form it has shape parameter k > 0 and scale parameter λ > 0. The probability density function is f(x) = (k/λ) (x/λ)^{k−1} exp[−(x/λ)^k] for x ≥ 0, and zero for x < 0. The cumulative distribution function is F(x) = 1 − exp[−(x/λ)^k], and the survival function is S(x) = exp[−(x/λ)^k].

The hazard rate is h(x) = f(x)/S(x) = (k/λ)(x/λ)^{k−1}. The mean is E[X] = λ Γ(1 + 1/k), and the variance is Var(X) = λ^2 [Γ(1 + 2/k) − Γ(1 + 1/k)^2]. The median is λ (ln 2)^{1/k}, and the mode is λ [ (k − 1)/k ]^{1/k} for k > 1. When k = 1 the Weibull reduces to the exponential distribution with mean λ.

Special cases and uses. For k = 2 it is related to the Rayleigh distribution (up to a scale factor). The distribution is widely used to model time to failure and reliability data, as well as in hydrology, climatology, and wind-speed analysis. Estimation of k and λ is typically done by maximum likelihood or by method of moments; closed-form solutions do not generally exist and numerical methods are used. The Weibull plot, a log–log plot of failure data, is often used to assess linearity and to estimate parameters.

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