Weibulljakaumat

Weibulljakaumat are a family of continuous probability distributions used to model time-to-failure and other lifespan data. Named after Waloddi Weibull, the distributions are characterized by two positive parameters, shape k and scale λ.

The cumulative distribution function is F(x) = 1 - exp(- (x/λ)^k) for x ≥ 0. The probability density function is f(x) = (k/λ) (x/λ)^{k-1} exp(- (x/λ)^k). The survival function is S(x) = exp(- (x/λ)^k), and the hazard function is h(x) = (k/λ) (x/λ)^{k-1}.

Moments of the distribution are E[X^n] = λ^n Γ(1 + n/k), with the mean E[X] = λ Γ(1 + 1/k) and the median m = λ (ln 2)^{1/k}. The shape parameter k controls the form of the hazard rate: k = 1 yields the exponential distribution, k > 1 indicates an increasing hazard with age, and k < 1 indicates a decreasing hazard.

Parameter estimation is typically performed through maximum likelihood, method of moments, or Bayesian methods. In reliability studies, censored data are common and MLE approaches accommodate them. Goodness of fit can be assessed using QQ plots, the Kolmogorov-Smirnov test, and information criteria such as AIC or BIC.

Applications of Weibulljakaumat are widespread in engineering and statistics. They are standard tools in reliability engineering and life data analysis for modeling time-to-failure, product lifetimes, and maintenance planning. They are also employed in fields such as hydrology, wind engineering, and material science, where flexible tail behavior helps capture diverse data patterns.