Weibull distribution

In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. Examples are maximum one-day rainfalls and the time a user spends on a web page. The distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939, although it was first identified by René Maurice Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.

Definition

Standard parameterization The probability density function of a Weibull random variable is

f ( x ; λ , k ) = { k λ ( x λ ) k − 1 e − ( x / λ ) k , x ≥ 0 , 0 , x < 0 , {\displaystyle f(x;\lambda ,k)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0,\end{cases}}}

where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2 and λ = 2 σ {\displaystyle \lambda ={\sqrt {2}}\sigma } ). If the quantity, x, is a "time-to-failure", the Weibull distribution gives a distribution for which the hazard rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:

A value of k < 1 {\displaystyle k<1\,} indicates that the hazard rate decreases over time (like in case of the Lindy effect, which however corresponds to Pareto distributions rather than Weibull distributions). This happens if there is significant "infant mortality", or defective items failing early and the hazard rate decreasing over time as the defective items are weeded out of the population. In the context of the diffusion of innovations, this means negative word of mouth: the hazard function is a monotonically decreasing function of the proportion of adopters; A value of k = 1 {\displaystyle k=1\,} indicates that the hazard rate is constant over time. This might suggest random external events are causing mortality, or failure. The Weibull distribution reduces to an exponential distribution; A value of k > 1 {\displaystyle k>1\,} indicates that the hazard rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the context of the diffusion of innovations, this means positive word of mouth: the hazard function is a monotonically increasing function of the proportion of adopters. The function is first convex, then concave with an inflection point at ( e 1 / k − 1 ) / e 1 / k , k > 1 {\displaystyle (e^{1/k}-1)/e^{1/k},\,k>1\,} . In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus. In the context of diffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model.

Optional parameterizations

First option Applications in medical statistics and econometrics often adopt a different parameterization. The shape parameter k is the same as above, while the scale parameter is b = λ − k {\displaystyle b=\lambda ^{-k}} . In this case, for x ≥ 0, the probability density function is

f ( x ; k , b ) = b k x k − 1 e − b x k , {\displaystyle f(x;k,b)=bkx^{k-1}e^{-bx^{k}},}

the cumulative distribution function is

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