View source for Weibull distribution

{{More footnotes|date=June 2010}}
{{Probability distribution
|name       =Weibull (2-parameter)
|type       =density
|pdf_image  =[[Image:Weibull PDF.svg|325px|Probability distribution function]]<!--- NOT CORRECT (see discussion) [[Image:Weibul pdf.png|325px|Probability distribution function]]--->
|cdf_image  =[[Image:Weibull CDF.svg|325px|Cumulative distribution function]]<!--- NOT CORRECT (see discussion) [[Image:Weibul cdf.png|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>\begin{cases}1- e^{-(x/\lambda)^k} & x\geq0\\ 0 & x<0\end{cases}</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|v|eɪ|b|ʊ|l}} is a continuous [[probability distribution]]. It is named after Swedish mathematician [[Waloddi Weibull]], who described it in detail in 1951, although it was first identified by {{harvtxt|Fréchet|1927}} and first applied by {{harvtxt|Rosin|Rammler|1933}} to describe a [[Particle-size distribution|particle size distribution]].

==Definition==

===Standard parameterization===

The [[probability density function]] of a Weibull [[random variable]] is:<ref>{{cite book |last=Papoulis |first=	Athanasios Papoulis |last2=Pillai |first2=S. Unnikrishna |title=Probability, Random Variables, and Stochastic Processes |location=Boston |publisher=McGraw-Hill |edition=4th |year=2002 |isbn=0-07-366011-6 }}</ref>

:<math>
f(x;\lambda,k) =
\begin{cases}
\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0 ,\\
0 & x<0,
\end{cases}</math>

where ''k'' > 0 is the ''[[shape parameter]]'' and λ > 0 is the ''[[scale parameter]]'' of the distribution. Its [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|complementary cumulative distribution function]] is a [[stretched exponential function]]. The Weibull distribution is related to a number of other probability distributions; in particular, it [[Interpolation|interpolates]] between the [[exponential distribution]] (''k'' = 1) and the [[Rayleigh distribution]] (''k'' = 2 and <math>\lambda = \sqrt{2}\sigma </math><ref>{{cite web|url=http://www.mathworks.com.au/help/stats/rayleigh-distribution.html|title=Rayleigh Distribution - MATLAB & Simulink - MathWorks Australia|author=|date=|website=www.mathworks.com.au}}</ref>).

If the quantity ''X'' is a "time-to-failure", the Weibull distribution gives a distribution for which the [[failure rate]] is proportional to a power of time. The ''shape'' parameter, ''k'', is that power plus one, and so this parameter can be interpreted directly as follows:<ref>{{cite journal |doi=10.1016/j.ress.2011.09.003 |title=A study of Weibull shape parameter: Properties and significance |journal=Reliability Engineering & System Safety |volume=96 |issue=12 |pages=1619–26 |year=2011 |last1=Jiang |first1=R. |last2=Murthy |first2=D.N.P. }}</ref>

* A value of <math> k < 1\,</math> indicates that the [[failure rate]] decreases over time ([[Lindy effect]]). This happens if there is significant "infant mortality", or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population. In the context of the [[diffusion of innovations]], this means negative word of mouth: the hazard function is a monotonically decreasing function of the proportion of adopters; 
* A value of <math> k = 1\,</math>  indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure. The Weibull distribution reduces to an exponential distribution; 
* A value of <math> k > 1\,</math> indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the context of the [[diffusion of innovations]], this means positive word of mouth: the hazard function is a monotonically increasing function of the proportion of adopters. The function is first convex, then concave with an inflexion point at <math>(e^{1/k} - 1)/e^{1/k}, k > 1\,</math>.

In the field of [[materials science]], the shape parameter ''k'' of a distribution of strengths is known as the [[Weibull modulus]]. In the context of [[diffusion of innovations]], the Weibull distribution is a "pure" imitation/rejection model.

===Alternative parameterizations===

In medical statistics a different parameterization is used.<ref>{{cite book |last=Collett |first=David |title=Modelling survival data in medical research |location=Boca Raton |publisher=Chapman and Hall / CRC |edition=3rd |year=2015 |isbn=978-1439856789 }}</ref> The shape parameter ''k'' is the same as above and the scale parameter is <math>b = \lambda^{-k}</math>. For ''x'' ≥ 0 the hazard function is
:<math>h(x;k,b) = bkx^{k-1},</math>

and the probability density function is 
:<math>f(x;k,b) = bkx^{k-1}\exp(-bx^k).</math>
The mean is 
:<math>b^{-1/k}\Gamma(1+1/k).</math>

A third parameterization is sometimes used. In this the shape parameter ''k'' is the same as above and the scale parameter is <math>\beta = 1/\lambda</math>.

==Properties==

===Density function===

The form of the density function of the Weibull distribution changes drastically with the value of ''k''. For 0 < ''k'' < 1, the density function tends to ∞ as ''x'' approaches zero from above and is strictly decreasing. For ''k'' = 1, the density function tends to 1/''λ'' as ''x'' approaches zero from above and is strictly decreasing. For ''k'' > 1, the density function tends to zero as ''x'' approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at ''x'' = 0 if 0 < ''k'' < 1, infinite positive slope at ''x'' = 0 if 1 < ''k'' < 2 and null slope at ''x'' = 0 if ''k'' > 2. For ''k'' = 1 the density has a finite negative slope at ''x'' = 0. For ''k'' = 2 the density has a finite positive slope at ''x'' = 0. As ''k'' goes to infinity, the Weibull distribution converges to a [[Dirac delta distribution]] centered at ''x'' = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter.

===Cumulative distribution function===

The [[cumulative distribution function]] for the Weibull distribution is

:<math>F(x;k,\lambda) = 1- e^{-(x/\lambda)^k}\,</math>

for ''x'' ≥ 0, and ''F''(''x''; ''k''; λ) = 0 for ''x'' < 0.

If ''x'' = λ then ''F''(''x''; ''k''; λ) =&nbsp;1&nbsp;−&nbsp;''e''<sup>−1</sup> ≈&nbsp;0.632 for all values of&nbsp;''k''. Vice versa: at ''F''(''x''; ''k''; ''λ'') = 0.632 the value of&nbsp;''x''&nbsp;≈&nbsp;''λ''.

The quantile (inverse cumulative distribution) function for the Weibull distribution is

:<math>Q(p;k,\lambda) = \lambda(-\ln(1-p))^{1/k}</math>

for 0 ≤ ''p'' < 1.

The [[failure rate]] ''h'' (or hazard function) is given by

:<math> h(x;k,\lambda) = {k \over \lambda} \left({x \over \lambda}\right)^{k-1}.</math>

===Moments===
The [[moment generating function]] of the [[logarithm]] of a Weibull distributed [[random variable]] is given by<ref name=JKB>{{harvnb|Johnson|Kotz|Balakrishnan|1994}}</ref>

:<math>\operatorname E\left[e^{t\log X}\right] = \lambda^t\Gamma\left(\frac{t}{k}+1\right)</math>

where {{math|Γ}} is the [[gamma function]]. Similarly, the [[characteristic function (probability theory)|characteristic function]] of log ''X'' is given by

:<math>\operatorname E\left[e^{it\log X}\right] = \lambda^{it}\Gamma\left(\frac{it}{k}+1\right).</math>

In particular, the ''n''th [[raw moment]] of ''X'' is given by

:<math>m_n = \lambda^n \Gamma\left(1+\frac{n}{k}\right).</math>

The [[mean]] and [[variance]] of a Weibull [[random variable]] can be expressed as

:<math>\operatorname{E}(X) = \lambda \Gamma\left(1+\frac{1}{k}\right)\,</math>

and

:<math>\operatorname{var}(X) = \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,.</math>

The skewness is given by

:<math>\gamma_1=\frac{\Gamma\left(1+\frac{3}{k}\right)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}</math>

where the mean is denoted by {{math|μ}} and the standard deviation is denoted by {{math|σ}}.

The excess [[kurtosis]] is given by

:<math>\gamma_2=\frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2
-4\Gamma_1 \Gamma_3 +\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2}</math>

where <math>\Gamma_i=\Gamma(1+i/k)</math>. The kurtosis excess may also be written as:

:<math>\gamma_{2}=\frac{\lambda^4\Gamma(1+\frac{4}{k})-4\gamma_{1}\sigma^3\mu-6\mu^2\sigma^2-\mu^4}{\sigma^4}-3</math>

===Moment generating function===
A variety of expressions are available for the moment generating function of ''X'' itself. As a [[power series]], since the raw moments are already known, one has

:<math>\operatorname E\left[e^{tX}\right] = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!} \Gamma\left(1+\frac{n}{k}\right).</math>

Alternatively, one can attempt to deal directly with the integral

:<math>\operatorname E\left[e^{tX}\right] = \int_0^\infty e^{tx} \frac k \lambda \left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}\,dx.</math>

If the parameter ''k'' is assumed to be a rational number, expressed as ''k'' = ''p''/''q'' where ''p'' and ''q'' are integers, then this integral can be evaluated analytically.<ref>See {{harv|Cheng|Tellambura|Beaulieu|2004}} for the case when ''k'' is an integer, and {{harv|Sagias|Karagiannidis|2005}} for the rational case.</ref> With ''t'' replaced by −''t'', one finds

:<math> \operatorname E\left[e^{-tX}\right] = \frac1{ \lambda^k\, t^k} \, \frac{ p^k \, \sqrt{q/p}} {(\sqrt{2 \pi})^{q+p-2}} \, G_{p,q}^{\,q,p} \!\left( \left. \begin{matrix} \frac{1-k}{p}, \frac{2-k}{p}, \dots, \frac{p-k}{p} \\ \frac{0}{q}, \frac{1}{q}, \dots, \frac{q-1}{q} \end{matrix} \; \right| \, \frac {p^p} {\left( q \, \lambda^k \, t^k \right)^q} \right) </math>
where ''G'' is the [[Meijer G-function]].

The [[characteristic function (probability theory)|characteristic function]] has also been obtained by {{harvtxt|Muraleedharan|Rao|Kurup|Nair|2007}}. The [[characteristic function (probability theory)|characteristic function]] and [[moment generating function]] of 3-parameter Weibull distribution have also been derived by {{harvtxt|Muraleedharan|Soares|2014}} by a direct approach.

===Shannon entropy===
The [[entropy (information theory)|information entropy]] is given by

:<math>
H(\lambda,k) = \gamma\left(1 - \frac{1}{k}\right) + \ln\left(\frac{\lambda}{k}\right) + 1
</math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. The Weibull distribution is the [[maximum entropy distribution]] for a non-negative real random variate with a fixed [[expected value]] of ''x''<sup>''k''</sup> equal to ''&lambda;''<sup>''k''</sup> and a fixed expected value of ln(''x''<sup>''k''</sup>) equal to  ln(''&lambda;''<sup>''k''</sup>)&nbsp;−&nbsp;<math>\gamma</math>.

===Parameter estimation===

====Maximum likelihood====

The [[Maximum likelihood estimation|maximum likelihood estimator]] for the <math>\lambda</math> parameter given <math>k</math> is

:<math>\widehat \lambda^k = \frac{1}{n} \sum_{i=1}^n x_i^k </math>

The maximum likelihood estimator for <math>k</math> is the solution for ''k'' of the following equation{{citation needed|date=December 2017}}
:<math>
  0 =  \frac{\sum_{i=1}^n x_i^k \ln x_i }
                       {\sum_{i=1}^n x_i^k }
                  - \frac{1}{k} - \frac{1}{n} \sum_{i=1}^n \ln x_i
</math>

This equation defining <math>\widehat k</math> only implicitly, one must generally solve for <math>k</math> by numerical means.

When <math>x_1 > x_2 > \cdots > x_N</math> are the <math>N</math> largest observed samples from a dataset of more than <math>N</math> samples,  then the maximum likelihood estimator for the <math>\lambda</math> parameter given <math>k</math> is<ref>{{cite book
 | author = Sornette, D.
 | year = 2004
 | title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref>

:<math>\widehat \lambda^k = \frac{1}{N} \sum_{i=1}^N (x_i^k - x_N^k)</math>

Also given that condition, the maximum likelihood estimator for <math>k</math> is{{citation needed|date=December 2017}}
:<math>
  0 = \frac{\sum_{i=1}^N (x_i^k \ln x_i -  x_N^k \ln x_N)}
                       {\sum_{i=1}^N (x_i^k - x_N^k)}
                  - \frac{1}{N} \sum_{i=1}^N \ln x_i
</math>

Again, this being an implicit function, one must generally solve for <math>k</math> by numerical means.

==Weibull plot==

The fit of data to a Weibull distribution can be visually assessed using a Weibull plot.<ref>{{cite web|url=http://www.itl.nist.gov/div898/handbook/eda/section3/weibplot.htm|title=1.3.3.30. Weibull Plot|author=|date=|website=www.itl.nist.gov}}</ref> The Weibull plot is a plot of the [[empirical cumulative distribution function]] <math>\widehat F(x)</math> of data on special axes in a type of [[Q-Q plot]]. The axes are <math>\ln(-\ln(1-\widehat F(x)))</math> versus <math>\ln(x)</math>. The reason for this change of variables is the cumulative distribution function can be linearized:
:<math>\begin{align}
F(x) &= 1-e^{-(x/\lambda)^k}\\[4pt]
-\ln(1-F(x)) &= (x/\lambda)^k\\[4pt]
\underbrace{\ln(-\ln(1-F(x)))}_{\textrm{'y'}} &= \underbrace{k\ln x}_{\textrm{'mx'}} - \underbrace{k\ln \lambda}_{\textrm{'c'}}
\end{align}
</math>
which can be seen to be in the standard form of a straight line. Therefore, if the data came from a Weibull distribution then a straight line is expected on a Weibull plot.

There are various approaches to obtaining the empirical distribution function from data: one method is to obtain the vertical coordinate for each point using <math>\widehat F = \frac{i-0.3}{n+0.4}</math> where <math>i</math> is the rank of the data point and <math>n</math> is the number of data points.<ref>Wayne Nelson (2004) ''Applied Life Data Analysis''. Wiley-Blackwell {{ISBN|0-471-64462-5}}</ref>

Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution. The gradient informs one directly about the shape parameter <math>k</math> and the scale parameter <math>\lambda</math> can also be inferred.

==[[Kullback–Leibler divergence]]==

: <math>D_\text{KL}( \mathrm{Weib}_1 \parallel \mathrm{Weib}_2) = \log \frac{k_1}{\lambda_1^{k_1}} - \log \frac{k_2}{\lambda_2^{k_2}} + (k_1 - k_2) \left[ \log \lambda_1 - \frac{\gamma}{k_1} \right] + \left(\frac{\lambda_1}{\lambda_2}\right)^{k_2}  \Gamma \left(\frac{k_2}{k_1} + 1 \right) - 1</math> <ref>{{Cite arXiv |eprint = 1310.3713|last1 = Bauckhage|first1 = Christian|title = Computing the Kullback-Leibler Divergence between two Weibull Distributions|class = cs.IT|year = 2013}}</ref>

==Applications==
The Weibull distribution is used{{Citation needed|date=June 2010}}

* In [[survival analysis]]<ref>{{cite web|url=http://www.statsoft.com/textbook/survival-failure-time-analysis/#distribution|title=Survival/Failure Time Analysis|last=www.statsoft.com|date=|website=www.statsoft.com}}</ref>
* In [[reliability engineering]] and [[failure analysis]]
* In [[electrical engineering]] to represent overvoltage occurring in an electrical system
* In [[industrial engineering]] to represent [[manufacturing]] and [[Delivery (commerce)|delivery]] times
* In [[extreme value theory]]
* In [[weather forecasting]] and the [[Wind power#Wind energy|wind power industry]] to describe [[Wind power#Distribution of wind speed|wind speed distributions]], as the natural distribution often matches the Weibull shape<ref>{{cite web|url=http://www.reuk.co.uk/Wind-Speed-Distribution-Weibull.htm|title=Wind Speed Distribution Weibull - REUK.co.uk|author=|date=|website=www.reuk.co.uk}}</ref>
* In communications systems engineering
** In [[radar]] systems to model the dispersion of the received signals level produced by some types of clutters
** To model [[fading channel]]s in [[wireless]] communications, as the [[Weibull fading]] model seems to exhibit good fit to experimental fading [[Channel (communications)|channel]] measurements
[[File:FitWeibullDistr.tif|thumb|240px|Fitted cumulative Weibull distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] <ref>{{cite web|url=https://www.waterlog.info/cumfreq.htm|title=CumFreq, Distribution fitting of probability, free software, cumulative frequency|author=|date=|website=}}</ref>  ]]

* In [[information retrieval]] to model dwell times on web pages<ref>{{Cite book|last=Liu|first=Chao|last2=White|first2=Ryen W.|last3=Dumais|first3=Susan|date=2010-07-19|title=Understanding web browsing behaviors through Weibull analysis of dwell time|publisher=ACM|pages=379–386|doi=10.1145/1835449.1835513|isbn=9781450301534}}</ref>.

* In [[general insurance]] to model the size of [[reinsurance]] claims, and the cumulative development of [[asbestosis]] losses
* In forecasting technological change (also known as the Sharif-Islam model)<ref>{{cite journal |doi=10.1016/0040-1625(80)90026-8 |title=The Weibull distribution as a general model for forecasting technological change |journal=Technological Forecasting and Social Change |volume=18 |issue=3 |pages=247–56 |year=1980 |last1=Sharif |first1=M.Nawaz |last2=Islam |first2=M.Nazrul }}</ref>
* In [[hydrology]] the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.
* In describing the size of [[Granular material|particles]] generated by grinding, [[mill (grinding)|milling]] and [[crusher|crushing]] operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin-Rammler distribution.{{Citation needed|date = May 2011}} In this context it predicts fewer fine particles than the [[Log-normal distribution]] and it is generally most accurate for narrow particle size distributions.{{Citation needed|date = May 2011}} The interpretation of the cumulative distribution function is that <math>F(x; k, \lambda)</math> is the [[Mass fraction (chemistry)|mass fraction]] of particles with diameter smaller than <math>x</math>, where <math>\lambda</math> is the mean particle size and <math>k</math> is a measure of the spread of particle sizes.
* In describing random point clouds (such as the positions of particles in an ideal gas): the probability to find the nearest-neighbor particle at a distance <math>x</math> from a given particle is given by a Weibull distribution with <math>k=3</math> and <math>\rho=1/\lambda^3</math> equal to the density of the particles.<ref>{{cite journal |last=Chandrashekar |first=S. |title=Stochastic Problems in Physics and Astronomy |journal=Reviews of Modern Physics |volume=15 |number=1 |year=1943 |page= 86}}</ref>

==Related distributions==
* {{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^{-({x-\theta \over \lambda})^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. When <math>\theta = 0</math>, this reduces to the 2-parameter distribution.<!--
        -->
* {{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"/><!--
        -->
* 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.<!--
        -->
* 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>.<!--
        -->
* 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 [[failure rate|rate]]s.<!--
        -->
* {{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><!--
        -->
* 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]].<!--
        -->
* {{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<!--
        -->
* 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 | url = | 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]].

==See also==
* [[Fisher–Tippett–Gnedenko theorem]]
* [[Logistic distribution]]
* [[Particle-size distribution#Rosin–Rammler distribution|Rosin–Rammler distribution]] for particle size analysis
* [[Rayleigh distribution]]

==References==
{{Reflist|30em}}

==Bibliography==
*{{citation|last=Fréchet|first=Maurice|authorlink=Maurice Fréchet|title=Sur la loi de probabilité de l'écart maximum|journal=Annales de la Société Polonaise de Mathematique, Cracovie|volume=6|pages=93–116|year=1927}}.
*{{Citation | last1=Johnson | first1=Norman L. | last2=Kotz | first2=Samuel | last3=Balakrishnan | first3=N. | title=Continuous univariate distributions. Vol. 1 | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | series=Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics | isbn=978-0-471-58495-7 | mr = 1299979| year=1994}}
*{{Citation | last1=Mann | first1=Nancy R. | author1-link=Nancy Mann | last2=Schafer | first2=Ray E. | last3=Singpurwalla | first3=Nozer D. | title=Methods for Statistical Analysis of Reliability and Life Data | publisher=[[John Wiley & Sons]] | location=New York | edition=1st | series=Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics | isbn=978-0-471-56737-0 | year=1974 | url-access=registration | url=https://archive.org/details/methodsforstatis00mann }}
*{{Citation | first1=G.|last1 = Muraleedharan | first2=A.D.|last2=Rao|first3=P.G.|last3=Kurup|first4=N. Unnikrishnan|last4=Nair|first5=Mourani|last5=Sinha| year = 2007 | title = Modified Weibull Distribution for Maximum and Significant Wave Height Simulation and Prediction | issue=8 | volume=54 |pages = 630–638|doi=10.1016/j.coastaleng.2007.05.001 | journal=Coastal Engineering}}
*{{citation|last1=Rosin|first1=P.|last2=Rammler|first2=E.|year=1933|title=The Laws Governing the Fineness of Powdered Coal|journal=Journal of the Institute of Fuel|volume=7|pages=29–36}}.
*{{cite journal |doi= 10.1109/TIT.2005.855598 |mr=2237527 |title=Gaussian Class Multivariate Weibull Distributions: Theory and Applications in Fading Channels |journal= IEEE Transactions on Information Theory |volume=51 |issue=10 |pages= 3608–19 |year=2005 |last1=Sagias |first1=N.C. |last2= Karagiannidis |first2= G.K.}}
*{{citation |last=Weibull |first=W. |authorlink= Waloddi Weibull |year=1951 |title=A statistical distribution function of wide applicability |journal= Journal of Applied Mechanics|volume=18 |issue=3 |pages= 293–297|url=http://www.barringer1.com/wa_files/Weibull-ASME-Paper-1951.pdf|bibcode=1951JAM....18..293W }}.
*{{cite book |chapter-url= http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm |chapter=Weibull Distribution |title= Engineering statistics handbook |publisher= [[National Institute of Standards and Technology]] |year=2008}}
*{{cite web |url= http://www.erpt.org/014Q/nelsa-06.htm |title=Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution |accessdate=2008-02-05 |last=Nelson, Jr |first=Ralph |date=2008-02-05}}

==External links==
* {{springer|title=Weibull distribution|id=p/w097370}}
* [http://www.mathpages.com/home/kmath122/kmath122.htm Mathpages – Weibull analysis]
* [http://reliawiki.org/index.php/The_Weibull_Distribution The Weibull Distribution]
* [http://www.crgraph.com/Weibull.pdf Reliability Analysis with Weibull]
* Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]
* [http://www.reliafy.com/ Online Weibull Probability Plotting]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Weibull Distribution}}
[[Category:Continuous distributions]]
[[Category:Survival analysis]]
[[Category:Exponential family distributions]]
[[Category:Extreme value data]]