Weibull distribution

In probability theory and statistics, the Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution. f ( x ; k , λ , θ ) = k λ ( x − θ λ ) k − 1 e − ( x − θ λ ) k {displaystyle f(x;k,lambda , heta )={k over lambda }left({x- heta over lambda } ight)^{k-1}e^{-({x- heta over lambda })^{k}},} X = ( W λ ) k {displaystyle X=left({frac {W}{lambda }} ight)^{k}} f F r e c h e t ( x ; k , λ ) = k λ ( x λ ) − 1 − k e − ( x / λ ) − k = − f W e i b u l l ( x ; − k , λ ) . {displaystyle f_{ m {Frechet}}(x;k,lambda )={frac {k}{lambda }}left({frac {x}{lambda }} ight)^{-1-k}e^{-(x/lambda )^{-k}}=-f_{ m {Weibull}}(x;-k,lambda ).} f ( x ; P 80 , m ) = { 1 − e ln ⁡ ( 0.2 ) ( x P 80 ) m x ≥ 0 , 0 x < 0 , {displaystyle f(x;P_{ m {80}},m)={egin{cases}1-e^{ln left(0.2 ight)left({frac {x}{P_{ m {80}}}} ight)^{m}}&xgeq 0,\0&x<0,end{cases}}} In probability theory and statistics, the Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution. The probability density function of a Weibull random variable is: 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 failure 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: 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. In medical statistics a different parameterization is used. The shape parameter k is the same as above and the scale parameter is b = λ − k {displaystyle b=lambda ^{-k}} . For x ≥ 0 the hazard function is