Weibull distribution information

Weibull (2-parameter)
	Probability density function
	Cumulative distribution function
Parameters	scale ; shape
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Excess kurtosis	(see text)
Entropy
MGF
CF
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.

^ Bowers, et. al. (1997) Actuarial Mathematics, 2nd ed. Society of Actuaries.

[1] Bowers, et. al. (1997) Actuarial Mathematics, 2nd ed. Society of Actuaries.

Weibull distribution information

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