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{{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]].

==Definition==

===Standard parameterization===

The [[probability density function]] of a Weibull [[random variable]] is<ref>{{cite book |last1=Papoulis |first1=	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><ref>{{cite journal |doi=10.1111/j.1740-9713.2018.01123.x |title=The Weibull distribution|journal=Significance |volume=15 |issue=2 |pages=10–11 |year=2018 |last1=Kizilersu |first1=Ayse |last2=Kreer |first2=Markus |last3=Thomas |first3=Anthony W. |doi-access=free }}</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|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 (like in case of the [[Lindy effect]], which however corresponds to [[Pareto distribution|Pareto distributions]]<ref name=":0">{{cite journal|last1=Eliazar|first1=Iddo|date=November 2017|title=Lindy's Law|journal=Physica A: Statistical Mechanics and Its Applications|volume=486|pages=797–805|bibcode=2017PhyA..486..797E|doi=10.1016/j.physa.2017.05.077|s2cid=125349686 }}</ref> rather than Weibull distributions). 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 [[Bass diffusion model|diffusion of innovations]], this means negative word of mouth: the [[Failure rate#hazard function|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 [[Bass diffusion model|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 inflection 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.

===Optional parameterizations===
====First option====
Applications in [[medical statistics]] and [[econometrics]] often adopt a different parameterization.<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><ref>{{cite book |last1=Cameron |first1=A. C. |last2=Trivedi |first2=P. K. |title=Microeconometrics : methods and applications |date=2005 |isbn=978-0-521-84805-3 |page=584|publisher=Cambridge University Press }}</ref> The shape parameter ''k'' is the same as above, while the scale parameter is <math>b = \lambda^{-k}</math>. In this case, for ''x'' ≥ 0, the probability density function is
:<math>f(x;k,b) = bkx^{k-1}e^{-bx^k},</math>
the cumulative distribution function is 
:<math>F(x;k,b) = 1 - e^{-bx^k},</math>
the quantile function is 
:<math>Q(p;k,b) = \left(-\frac{1}{b}\ln(1-p) \right)^{\frac{1}{k}},</math>
the hazard function is
:<math>h(x;k,b) = bkx^{k-1},</math>

and the mean is 
:<math>b^{-1/k}\Gamma(1+1/k).</math>
====Second option====
A second parameterization option can also be found.<ref>{{Cite book|last1=Kalbfleisch|first1=J. D.|title=The statistical analysis of failure time data|last2=Prentice|first2=R. L.|publisher=J. Wiley|year=2002|isbn=978-0-471-36357-6|edition=2nd|location=Hoboken, N.J.|oclc=50124320}}</ref><ref>{{Cite web|last=Therneau|first=T.|date=2020|others=R package version 3.1.|title=A Package for Survival Analysis in R.|url=https://CRAN.R-project.org/package=survival}}</ref> The shape parameter ''k'' is the same as in the standard case, while the scale parameter ''&lambda;'' is replaced with a rate parameter ''&beta;'' = 1/''&lambda;''. Then, for ''x'' ≥ 0, the probability density function is
:<math>f(x;k,\beta) = \beta k({\beta x})^{k-1} e^{-(\beta x)^k}</math>
the cumulative distribution function is 
:<math>F(x;k,\beta) = 1 - e^{-(\beta x)^k},</math>
the quantile function is 
:<math>Q(p;k,\beta) = \frac{1}{\beta}(-\ln(1-p))^\frac{1}{k},</math>
and the hazard function is
:<math>h(x;k,\beta) = \beta k({\beta x})^{k-1}.</math>

In all three parameterizations, the hazard is decreasing for k < 1, increasing for k > 1 and constant for k = 1, in which case the Weibull distribution reduces to an exponential distribution.

==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. A generalization of the Weibull distribution is the [[hyperbolastic functions|hyperbolastic distribution of type III]].

===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>

The [[Mean time between failures]] ''MTBF'' is

:<math> \text{MTBF}(k,\lambda) = \lambda\Gamma(1+1/k).</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{2\Gamma_1^3-3\Gamma_1\Gamma_2+ \Gamma_3 }{[\Gamma_2-\Gamma_1^2]^{3/2}}</math>

where <math>\Gamma_i=\Gamma(1+i/k)</math>, which may also be written as  

:<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.{{efn |See {{harv|Cheng|Tellambura|Beaulieu|2004}} for the case when ''k'' is an integer, and {{harv|Sagias|Karagiannidis|2005}} for the rational case.}} 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.

===Minima===
Let <math>X_1, X_2, \ldots, X_n</math> be independent and identically distributed Weibull random variables with scale parameter <math>\lambda</math> and shape parameter <math>k</math>. If the minimum of these <math>n</math> random variables is <math>Z = \min(X_1, X_2, \ldots, X_n)</math>, then the cumulative probability distribution of <math>Z</math> is given by 

:<math>F(z) = 1 - e^{-n(z/\lambda)^k}.</math>

That is, <math>Z</math> will also be Weibull distributed with scale parameter <math>n^{-1/k} \lambda</math> and with shape parameter <math>k</math>.

=== Reparametrization tricks ===
Fix some <math>\alpha > 0</math>. Let <math>(\pi_1, ..., \pi_n)</math> be nonnegative, and not all zero, and let <math>g_1,... , g_n</math> be independent samples of <math>\text{Weibull}(1, \alpha^{-1})</math>, then<ref>{{Cite journal |last1=Balog |first1=Matej |last2=Tripuraneni |first2=Nilesh |last3=Ghahramani |first3=Zoubin |last4=Weller |first4=Adrian |date=2017-07-17 |title=Lost Relatives of the Gumbel Trick |url=https://proceedings.mlr.press/v70/balog17a.html |journal=International Conference on Machine Learning |language=en |publisher=PMLR |pages=371–379|arxiv=1706.04161 }}</ref>

* <math>\arg\min_i (g_i \pi_i^{-\alpha}) \sim \text{Categorical}\left(\frac{\pi_j}{\sum_i \pi_i}\right)_j</math>
* <math>\min_i (g_i \pi_i^{-\alpha}) \sim\text{Weibull}\left( \left(\sum_i \pi_i \right)^{-\alpha}, \alpha^{-1}\right)</math>.

===Shannon entropy===
The [[entropy (information theory)|information entropy]] is given by<ref>{{Cite journal |last1=Cho |first1=Youngseuk |last2=Sun |first2=Hokeun |last3=Lee |first3=Kyeongjun |date=5 January 2015 |title=Estimating the Entropy of a Weibull Distribution under Generalized Progressive Hybrid Censoring |journal=Entropy |language=en |volume=17 |issue=1 |pages=102–122 |doi=10.3390/e17010102 |doi-access=free |bibcode=2015Entrp..17..102C |issn=1099-4300}}</ref>

:<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>.

=== Kullback–Leibler divergence ===
The [[Kullback–Leibler divergence]] between two Weibull distributions is given by<ref>{{Cite arXiv |eprint=1310.3713 |class=cs.IT |first1=Christian |last1=Bauckhage |title=Computing the Kullback-Leibler Divergence between two Weibull Distributions |year=2013}}</ref>
: <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>

==Parameter estimation==
===Ordinary least square using Weibull plot===
[[File:Weibull qq.svg|thumb|right|Weibull plot]]
The fit of a Weibull distribution to data 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|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><ref>{{Cite journal |last=Barnett |first=V. |date=1975 |title=Probability Plotting Methods and Order Statistics |url=https://www.jstor.org/stable/2346708 |journal=Journal of the Royal Statistical Society. Series C (Applied Statistics) |volume=24 |issue=1 |pages=95–108 |doi=10.2307/2346708 |jstor=2346708 |issn=0035-9254}}</ref> Another common estimator<ref>{{Cite ISO standard | csnumber = 69875 | title = ISO 20501:2019 – Fine ceramics (advanced ceramics, advanced technical ceramics) – Weibull statistics for strength data}}</ref> is 
:<math>\widehat F = \frac{i-0.5}{n}</math>.

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.

===Method of moments===
The [[coefficient of variation]] of Weibull distribution depends only on the shape parameter:<ref name="Cohen1965">{{cite journal | url=https://www.stat.cmu.edu/technometrics/59-69/VOL-07-04/v0704579.pdf |title=Maximum Likelihood Estimation in the Weibull Distribution Based on Complete and on Censored Samples |first=A. Clifford |last=Cohen | journal=Technometrics |issue=4 |volume=7 |date=Nov 1965 | pages=579–588|doi=10.1080/00401706.1965.10490300 }}</ref> 

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

Equating the sample quantities <math>s^2/\bar{x}^2</math> to <math>\sigma^2/\mu^2</math>, the moment estimate of the shape parameter <math>k</math> can be read off either from a look up table or a graph of <math>CV^2</math> versus <math>k</math>. A more accurate estimate of <math>\hat{k}</math> can be found using a root finding algorithm to solve

:<math>\frac{\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2}{\left(\Gamma\left(1+\frac{1}{k}\right)\right)^2} = \frac{s^2}{\bar{x}^2}.</math>

The moment estimate of the scale parameter can then be found using the first moment equation as 

:<math>\hat{\lambda} = \frac{\bar{x}}{\Gamma\left(1 + \frac{1}{\hat{k}}\right)}.</math>

===Maximum likelihood===
The [[Maximum likelihood estimation|maximum likelihood estimator]] for the <math>\lambda</math> parameter given <math>k</math> is<ref name="Cohen1965"/>

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

The maximum likelihood estimator for <math>k</math> is the solution for ''k'' of the following equation<ref name="Sornette, D. 2004">{{cite book
 | author = Sornette, D.
 | year = 2004
 | title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref>
:<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 defines <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 name="Sornette, D. 2004"/>

:<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.

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

[[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}}</ref>  ]]
[[File:DCA with four RDC.png|thumb|240px|Fitted curves for oil production time series data<ref name="ReferenceA">{{Cite journal |last1=Lee|first1=Se Yoon |first2=Bani|last2=Mallick|  title = Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas|journal=Sankhya B|year=2021|volume=84 |pages=1–43 |doi=10.1007/s13571-020-00245-8|doi-access=free}}</ref>]]

* In [[survival analysis]]
* 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 resources|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|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
* In [[information retrieval]] to model dwell times on web pages.<ref>{{Cite book|last1=Liu|first1=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|s2cid=12186028 }}</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 [[decline curve analysis]] to model oil production rate curve of shale oil wells.<ref name="ReferenceA"/>
* 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.<ref>[https://books.google.com/books?id=9RFdUPgpysEC&dq=Rosin-Rammler+distribution+2+Parameter+Weibull+distribution&pg=PA49 Computational Optimization of Internal Combustion Engine] page 49</ref> In this context it predicts fewer fine particles than the [[log-normal distribution]] and it is generally most accurate for narrow particle size distributions.<ref>{{cite book |last1=Austin |first1=L. G. |last2=Klimpel |first2=R. R. |last3=Luckie |first3=P. T. |title=Process Engineering of Size Reduction |date=1984 |publisher=Guinn Printing Inc. |location=Hoboken, NJ |isbn=0-89520-421-5}}</ref> 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|doi=10.1103/RevModPhys.15.1 |bibcode=1943RvMP...15....1C }}</ref>
* In calculating the rate of radiation-induced [[Radiation hardening#Digital_damage:_SEE|single event effects]] onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured device [[Cross section (physics)|cross section probability]] data to a particle [[linear energy transfer]] spectrum.<ref>{{cite report|date= November 15, 2008|title= ECSS-E-ST-10-12C – Methods for the calculation of radiation received and its effects, and a policy for design margins|url= https://ecss.nl/standard/ecss-e-st-10-12c-methods-for-the-calculation-of-radiation-received-and-its-effects-and-a-policy-for-design-margins/|publisher= European Cooperation for Space Standardization}}</ref> The Weibull fit was originally used because of a belief that particle energy levels align to a statistical distribution, but this belief was later proven false{{cn|date=November 2023}} and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.<ref>{{cite report|author1=L. D. Edmonds|author2= C. E. Barnes|author3= L. Z. Scheick|date= May 2000|title= An Introduction to Space Radiation Effects on Microelectronics|url= https://parts.jpl.nasa.gov/pdf/JPL00-62.pdf|publisher= NASA Jet Propulsion Laboratory, California Institute of Technology|section= 8.3 Curve Fitting|pages=75–76}}</ref>

==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"/><!--
        -->
* 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><!--
        -->
* 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<!--
        -->
* 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]].

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

==Notes==
{{Notelist}}

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

==Sources==
{{Refbegin |30em |indent=yes}}
*{{citation |last1=Cheng |last2=Tellambura |last3=Beaulieu |year=2004 |title=((n.t.)) }}{{full citation needed |date=April 2025}}
*{{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 | 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|bibcode = 2007CoasE..54..630M }}
*{{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.|s2cid=14654176 }}
*{{citation |last1=Muraleedharan |last2=Soares |year=2014 |title=((n.t.)) }}{{full citation needed |date=April 2025}}
*{{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}}
{{Refend}}

==Further reading==
{{Refbegin |30em |indent=yes}}
*{{citation|last=Fréchet|first=Maurice|author-link=Maurice Fréchet|title=Sur la loi de probabilité de l'écart maximum|journal=Annales de la Société Polonaise de Mathématique, Cracovie|volume=6|pages=93–116|year=1927}}.
*{{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 |last=Weibull |first=W. |author-link= Waloddi Weibull |year=1951 |title=A statistical distribution function of wide applicability |journal= Journal of Applied Mechanics|volume=18 |issue=3 |pages= 293–297|doi=10.1115/1.4010337 |url=http://web.cecs.pdx.edu/~cgshirl/Documents/Weibull-ASME-Paper-1951.pdf|bibcode=1951JAM....18..293W }}.
*{{cite web |url= http://www.erpt.org/014Q/nelsa-06.htm |title=Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution |access-date=2008-02-05 |last=Nelson Jr |first=Ralph |date=2008-02-05}}
{{Refend}}

==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}}
{{Authority control}}

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