Editing Weibull distribution (section)

==Related distributions==
* If <math>W \sim \mathrm{Weibull}(\lambda, k)</math>, then the variable <math>G = \log W</math> is Gumbel (minimum) distributed with location parameter <math>\mu = \log \lambda</math> and scale parameter <math>\beta = 1/k</math>. That is, <math>G \sim \mathrm{Gumbel}_{\min}(\log \lambda, 1/k)</math>.
* {{paragraph break}}A Weibull distribution is a [[generalized gamma distribution]] with both shape parameters equal to ''k''.<!--
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* {{paragraph break}}The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter.<ref name="JKB"/> It has the [[probability density function]] <blockquote><math>f(x;k,\lambda, \theta)={k \over \lambda} \left({x - \theta \over \lambda}\right)^{k-1} e^{-\left({x-\theta \over \lambda}\right)^k}\,</math></blockquote> {{paragraph break}}for <math>x \geq \theta</math> and <math>f(x; k, \lambda, \theta) = 0</math> for <math>x < \theta</math>, where <math>k > 0</math> is the [[shape parameter]], <math>\lambda > 0</math> is the [[scale parameter]] and <math>\theta</math> is the [[location parameter]] of the distribution. <math>\theta</math> value sets an initial failure-free time before the regular Weibull process begins. When <math>\theta = 0</math>, this reduces to the 2-parameter distribution.<!--
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* {{paragraph break}}The Weibull distribution can be characterized as the distribution of a random variable <math>W</math> such that the random variable <blockquote><math>X = \left(\frac{W}{\lambda}\right)^k</math></blockquote> {{paragraph break}}is the standard [[exponential distribution]] with intensity 1.<ref name="JKB"/><!--
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* This implies that the Weibull distribution can also be characterized in terms of a [[Uniform distribution (continuous)|uniform distribution]]: if <math>U</math> is uniformly distributed on <math>(0,1)</math>, then the random variable <math>W = \lambda(-\ln(U))^{1/k}\,</math> is Weibull distributed with parameters <math>k</math> and <math>\lambda</math>. Note that <math>-\ln(U)</math> here is equivalent to <math>X</math> just above. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.<!--
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* The Weibull distribution interpolates between the exponential distribution with intensity <math>1/\lambda</math> when <math>k = 1</math> and a [[Rayleigh distribution]] of mode <math>\sigma = \lambda/\sqrt{2}</math> when <math>k = 2</math>.<!--
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* The Weibull distribution (usually sufficient in [[reliability engineering]]) is a special case of the three parameter [[exponentiated Weibull distribution]] where the additional exponent equals 1. The exponentiated Weibull distribution accommodates [[Unimodal function|unimodal]], [[Bathtub curve|bathtub shaped]]<ref>{{cite web|url=http://www.sys-ev.com/reliability01.htm|title=System evolution and reliability of systems|publisher=Sysev (Belgium)|date=2010-01-01}}</ref> and [[Monotonic function|monotone]] [[failure rate]]s.<!--
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* {{paragraph break}}The Weibull distribution is a special case of the [[generalized extreme value distribution]]. It was in this connection that the distribution was first identified by [[Maurice Fréchet]] in 1927.<ref>{{cite book|last=Montgomery|first=Douglas|title=Introduction to statistical quality control|publisher=John Wiley|location=[S.l.]|isbn=9781118146811|page=95|date=2012-06-19}}</ref> The closely related [[Fréchet distribution]], named for this work, has the probability density function <blockquote><math>f_{\rm{Frechet}}(x;k,\lambda)=\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{-1-k} e^{-(x/\lambda)^{-k}} = f_{\rm{Weibull}}(x;-k,\lambda).</math></blockquote><!--
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* The distribution of a random variable that is defined as the minimum of several random variables, each having a different Weibull distribution, is a [[poly-Weibull distribution]].<!--
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* {{paragraph break}}The Weibull distribution was first applied by {{harvtxt|Rosin|Rammler|1933}} to describe particle size distributions. It is widely used in [[mineral processing]] to describe [[particle size distribution]]s in [[comminution]] processes. In this context the cumulative distribution is given by <blockquote><math>f(x;P_{\rm{80}},m) =  \begin{cases}
1-e^{\ln\left(0.2\right)\left(\frac{x}{P_{\rm{80}}}\right)^m} & x\geq0 ,\\
0 & x<0 ,\end{cases}</math></blockquote> {{paragraph break}}where
** <math>x</math> is the particle size
** <math>P_{\rm{80}}</math> is the 80th percentile of the particle size distribution
** <math>m</math> is a parameter describing the spread of the distribution<!--
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* Because of its availability in [[spreadsheet]]s, it is also used where the underlying behavior is actually better modeled by an [[Erlang distribution]].<ref>{{cite journal | last1 = Chatfield | first1 = C. | last2 = Goodhardt | first2 = G.J. | year = 1973 | title = A Consumer Purchasing Model with Erlang Interpurchase Times | journal = Journal of the American Statistical Association | volume = 68 | issue = 344| pages = 828–835 | doi=10.1080/01621459.1973.10481432}}</ref>
* If <math>X \sim \mathrm{Weibull}(\lambda,\frac{1}{2})</math> then <math> \sqrt{X} \sim \mathrm{Exponential}(\frac{1}{\sqrt{\lambda}})</math> ([[Exponential distribution]])
* {{paragraph break}}For the same values of k, the [[Gamma distribution]] takes on similar shapes, but the Weibull distribution is more [[Kurtosis#Excess kurtosis|platykurtic]].

* {{paragraph break}} From the viewpoint of the [[Stable count distribution]], <math> k </math> can be regarded as Lévy's stability parameter. A Weibull distribution can be decomposed to an integral of kernel density where the kernel is either a [[Laplace distribution]] <math>F(x;1,\lambda)</math> or a [[Rayleigh distribution]] <math>F(x;2,\lambda)</math>: <blockquote><math>     F(x;k,\lambda) =       \begin{cases}         \displaystyle\int_0^\infty \frac{1}{\nu} \, F(x;1,\lambda\nu)          \left( \Gamma \left( \frac{1}{k}+1 \right) \mathfrak{N}_k(\nu) \right) \, d\nu ,       & 1 \geq k > 0; \text{or } \\         \displaystyle\int_0^\infty \frac{1}{s} \, F(x;2,\sqrt{2} \lambda s)           \left( \sqrt{\frac{2}{\pi}} \, \Gamma \left( \frac{1}{k}+1 \right) V_k(s) \right) \, ds ,         & 2 \geq k > 0;       \end{cases} </math></blockquote> {{paragraph break}}where <math>\mathfrak{N}_k(\nu)</math> is the [[Stable count distribution]] and <math>V_k(s)</math> is the [[Stable_count_distribution#Stable_Vol_Distribution|Stable vol distribution]].