Editing Weibull distribution (section)

==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.

===Alternative parameterizations===
====First alternative====
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 alternative====
A second alternative parameterization 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.