Weibull distribution

In probability theory and statistics, the Weibull distribution /ˈwaɪbʊl/ 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.

Weibull (2-parameter)

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \lambda \in (0,+\infty )\,}$ scale ${\displaystyle k\in (0,+\infty )\,}$ shape
Support	${\displaystyle x\in [0,+\infty )\,}$
PDF	${\displaystyle f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0.\end{cases}}}$
CDF	${\displaystyle {\begin{cases}1-e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0.\end{cases}}}$
Mean	${\displaystyle \lambda \,\Gamma (1+1/k)\,}$
Median	${\displaystyle \lambda (\ln 2)^{1/k}\,}$
Mode	${\displaystyle {\begin{cases}\lambda \left({\frac {k-1}{k}}\right)^{1/k}\,,&k>1,\\0,&k\leq 1.\end{cases}}}$
Variance	${\displaystyle \lambda ^{2}\left[\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}\right]\,}$
Skewness	${\displaystyle {\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}}$
Ex. kurtosis	(see text)
Entropy	${\displaystyle \gamma (1-1/k)+\ln(\lambda /k)+1\,}$
MGF	${\displaystyle \sum _{n=0}^{\infty }{\frac {t^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k),\ k\geq 1}$
CF	${\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k)}$
Kullback–Leibler divergence	see below

The distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939, although it was first identified by Maurice René Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.