Weibull distribution

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	${\displaystyle F(x)={\begin{cases}1-e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0.\end{cases}}}$
Quantile	${\displaystyle Q(p)=\lambda (-\ln(1-p))^{\frac {1}{k}}}$
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	${\displaystyle {\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}}$
Excess 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

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.

The distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939,^[1] although it was first identified by René Maurice Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.