Weibullmodel

The Weibull model is a continuous probability distribution used extensively in reliability engineering and survival analysis. It is named after the Swedish mathematician Waloddi Weibull and was first presented in 1939. The model is particularly well-suited for describing phenomena that exhibit wear-out, infant mortality, or a constant failure rate over time.

The Weibull distribution is characterized by two parameters: the shape parameter, often denoted by k or $\beta$, and the scale parameter, often denoted by $\lambda$ or $\eta$. The shape parameter governs the form of the distribution. If k < 1, the failure rate decreases over time, representing infant mortality. If k = 1, the failure rate is constant, similar to the exponential distribution. If k > 1, the failure rate increases over time, indicating wear-out. The scale parameter influences the spread or time at which the failures occur.

A third parameter, the location parameter or shift parameter ($\gamma$), can also be included, which shifts the distribution along the time axis. This is useful when failures cannot begin before a certain time. The probability density function (PDF) of the Weibull distribution is given by $f(x; k, \lambda) = (k/\lambda)(x/\lambda)^{k-1}e^{-(x/\lambda)^k}$ for x $\ge$ 0, and 0 otherwise. The cumulative distribution function (CDF) is $F(x; k, \lambda) = 1 - e^{-(x/\lambda)^k}$.

Due to its flexibility in modeling various failure patterns, the Weibull model is widely applied in industries such as aerospace, manufacturing, and medicine for predicting product lifetimes, analyzing system reliability, and assessing the effectiveness of treatments.