Editing Weibull distribution (section)

{{Short description|Continuous probability distribution}}
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{{Infobox probability distribution
|name       =Weibull (2-parameter)
|type       =density
|pdf_image  =[[Image:Weibull PDF.svg|325px|Probability distribution function]]
|cdf_image  =[[Image:Weibull CDF.svg|325px|Cumulative distribution function]]
|parameters =<math>\lambda\in (0, +\infty)\,</math> [[scale parameter|scale]] <br/><math>k\in (0, +\infty)\,</math> [[shape parameter|shape]]
|support    =<math>x \in [0, +\infty)\,</math>
|pdf        =<math>f(x)=\begin{cases}
\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}, & x\geq0,\\
0, & x<0.\end{cases}</math>
|cdf        =<math>F(x)=\begin{cases}1 - e^{-(x/\lambda)^k}, & x\geq0,\\ 0, & x<0.\end{cases}</math>
|quantile   =<math>Q(p)=\lambda(-\ln(1-p))^\frac{1}{k}</math>
|mean       =<math>\lambda \, \Gamma(1+1/k)\,</math>
|median     =<math>\lambda(\ln2)^{1/k}\,</math>
|mode       =<math>\begin{cases}
\lambda \left(\frac{k-1}{k} \right)^{1/k}\,, &k>1,\\
0, &k\leq 1.\end{cases}</math>

|variance   =<math>\lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,</math>
|skewness   =<math>\frac{\Gamma(1+3/k)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}</math>
|kurtosis   =(see text)
|entropy    =<math>\gamma(1-1/k)+\ln(\lambda/k)+1 \,</math>
|mgf        = <math>\sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma(1+n/k), \ k\geq1</math>
|char       = <math>\sum_{n=0}^\infty \frac{(it)^n\lambda^n}{n!}\Gamma(1+n/k)</math>
|KLDiv      = see below 
}}

In [[probability theory]] and [[statistics]], the '''Weibull distribution''' {{IPAc-en|ˈ|w|aɪ|b|ʊ|l}} 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,<ref>{{cite journal| author=W. Weibull| date=1939| title=The Statistical Theory of the Strength of Materials| language=en| journal=Ingeniors Vetenskaps Academy Handlingar| issue=151 | publisher=Generalstabens Litografiska Anstalts Förlag| location=Stockholm| pages=1–45}}</ref><ref>Bowers, et. al. (1997) Actuarial Mathematics, 2nd ed. Society of Actuaries.</ref> although it was first identified by [[René Maurice Fréchet]] and first applied by {{harvtxt|Rosin|Rammler|1933}} to describe a [[Particle-size distribution|particle size distribution]].