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Weibull (2-parameter)

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \lambda \in (0,+\mathrm{\infty })}$scale ${\displaystyle k\in (0,+\mathrm{\infty })}$shape
Support	${\displaystyle x\in [0,+\mathrm{\infty })}$
PDF
CDF
Quantile	${\displaystyle Q(p)=\lambda (-\ln (1-p){)}^{\frac{1}{k}}}$
Mean	${\displaystyle \lambda \mathrm{\Gamma }(1+1/k)}$
Median	${\displaystyle \lambda (\ln 2{)}^{1/k}}$
Mode
Variance	${\displaystyle {\lambda }^2\left[\mathrm{\Gamma }\left(1+\frac{2}{k}\right)-{\left(\mathrm{\Gamma }\left(1+\frac{1}{k}\right)\right)}^2\right]}$
Skewness	${\displaystyle \frac{\mathrm{\Gamma }(1+3/k){\lambda }^3-3\mu {\sigma }^2-{\mu }^3}{{\sigma }^3}}$
Excess kurtosis	(see text)
Entropy	${\displaystyle \gamma (1-1/k)+\ln (\lambda /k)+1}$
MGF	${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{t^n{\lambda }^n}{n!}\mathrm{\Gamma }(1+n/k),k\ge 1}$
CF	${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{(it{)}^n{\lambda }^n}{n!}\mathrm{\Gamma }(1+n/k)}$
Kullback–Leibler divergence	see below

Inprobability theory andstatistics, theWeibull distribution/ˈwaɪbʊl/ is a continuousprobability 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 mathematicianWaloddi Weibull, who described it in detail in 1939,^[1][2] although it was first identified byRené Maurice Fréchet and first applied byRosin & Rammler (1933) to describe aparticle size distribution.

Theprobability density function of a Weibullrandom variable is^[3][4]

wherek > 0 is theshape parameter and λ > 0 is thescale parameter of the distribution. Itscomplementary cumulative distribution function is astretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, itinterpolates between theexponential distribution (k = 1) and theRayleigh distribution (k = 2 and${\displaystyle \lambda =\sqrt{2}\sigma }$).^[5]

If the quantity,x, is a "time-to-failure", the Weibull distribution gives a distribution for which thefailure rate is proportional to a power of time. Theshape parameter,k, is that power plus one, and so this parameter can be interpreted directly as follows:^[6]

• A value of indicates that thefailure rate decreases over time (like in case of theLindy effect, which however corresponds toPareto distributions^[7] 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 thediffusion of innovations, this means negative word of mouth: thehazard function is a monotonically decreasing function of the proportion of adopters;
• A value of${\displaystyle k=1}$ 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${\displaystyle k>1}$ 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 thediffusion 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${\displaystyle (e^{1/k}-1)/e^{1/k},k>1}$.

In the field ofmaterials science, the shape parameterk of a distribution of strengths is known as theWeibull modulus. In the context ofdiffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model.

Applications inmedical statistics andeconometrics often adopt a different parameterization.^[8][9] The shape parameterk is the same as above, while the scale parameter is${\displaystyle b={\lambda }^{-k}}$. In this case, forx ≥ 0, the probability density function is

${\displaystyle f(x;k,b)=bk{x}^{k-1}{e}^{-b{x}^k},}$

the cumulative distribution function is

${\displaystyle F(x;k,b)=1-e^{-b{x}^k},}$

the quantile function is

${\displaystyle Q(p;k,b)={\left(-\frac{1}{b}\ln (1-p)\right)}^{\frac{1}{k}},}$

the hazard function is

${\displaystyle h(x;k,b)=bk{x}^{k-1},}$

and the mean is

${\displaystyle b^{-1/k}\mathrm{\Gamma }(1+1/k).}$

A second alternative parameterization can also be found.^[10][11] The shape parameterk is the same as in the standard case, while the scale parameterλ is replaced with a rate parameterβ = 1/λ. Then, forx ≥ 0, the probability density function is

${\displaystyle f(x;k,\beta )=\beta k(\beta x{)}^{k-1}{e}^{-(\beta x{)}^k}}$

the cumulative distribution function is

${\displaystyle F(x;k,\beta )=1-e^{-(\beta x{)}^k},}$

the quantile function is

${\displaystyle Q(p;k,\beta )=\frac{1}{\beta }(-\ln (1-p){)}^{\frac{1}{k}},}$

and the hazard function is

${\displaystyle h(x;k,\beta )=\beta k(\beta x{)}^{k-1}.}$

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.

The form of the density function of the Weibull distribution changes drastically with the value ofk. For 0 <k < 1, the density function tends to ∞ asx approaches zero from above and is strictly decreasing. Fork = 1, the density function tends to 1/λ asx approaches zero from above and is strictly decreasing. Fork > 1, the density function tends to zero asx approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope atx = 0 if 0 <k < 1, infinite positive slope atx = 0 if 1 <k < 2 and null slope atx = 0 ifk > 2. Fork = 1 the density has a finite negative slope atx = 0. Fork = 2 the density has a finite positive slope atx = 0. Ask goes to infinity, the Weibull distribution converges to aDirac delta distribution centered atx = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter. A generalization of the Weibull distribution is thehyperbolastic distribution of type III.

Thecumulative distribution function for the Weibull distribution is

${\displaystyle F(x;k,\lambda )=1-e^{-(x/\lambda {)}^k}}$

forx ≥ 0, andF(x;k; λ) = 0 forx < 0.

Ifx = λ thenF(x;k; λ) = 1 − e⁻¹ ≈ 0.632 for all values of k. Vice versa: atF(x;k;λ) = 0.632 the value of x ≈ λ.

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

${\displaystyle Q(p;k,\lambda )=\lambda (-\ln (1-p){)}^{1/k}}$

for 0 ≤p < 1.

Thefailure rateh (or hazard function) is given by

${\displaystyle h(x;k,\lambda )=\frac{k}{\lambda }{\left(\frac{x}{\lambda }\right)}^{k-1}.}$

TheMean time between failuresMTBF is

${\displaystyle \text{MTBF}(k,\lambda )=\lambda \mathrm{\Gamma }(1+1/k).}$

Themoment generating function of thelogarithm of a Weibull distributedrandom variable is given by^[12]

${\displaystyle \mathrm{E}\left[e^{t\log X}\right]={\lambda }^t\mathrm{\Gamma }\left(\frac{t}{k}+1\right)}$

whereΓ is thegamma function. Similarly, thecharacteristic function of logX is given by

${\displaystyle \mathrm{E}\left[e^{it\log X}\right]={\lambda }^{it}\mathrm{\Gamma }\left(\frac{it}{k}+1\right).}$

In particular, thenthraw moment ofX is given by

${\displaystyle m_n={\lambda }^n\mathrm{\Gamma }\left(1+\frac{n}{k}\right).}$

Themean andvariance of a Weibullrandom variable can be expressed as

${\displaystyle \mathrm{E}(X)=\lambda \mathrm{\Gamma }\left(1+\frac{1}{k}\right)}$

and

${\displaystyle \mathrm{var}(X)={\lambda }^2\left[\mathrm{\Gamma }\left(1+\frac{2}{k}\right)-{\left(\mathrm{\Gamma }\left(1+\frac{1}{k}\right)\right)}^2\right].}$

The skewness is given by

${\displaystyle {\gamma }_1=\frac{2{\mathrm{\Gamma }}_1^3-3{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3}{[{\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1^2{]}^{3/2}}}$

where${\displaystyle {\mathrm{\Gamma }}_i=\mathrm{\Gamma }(1+i/k)}$, which may also be written as

${\displaystyle {\gamma }_1=\frac{\mathrm{\Gamma }\left(1+\frac{3}{k}\right){\lambda }^3-3\mu {\sigma }^2-{\mu }^3}{{\sigma }^3}}$

where the mean is denoted byμ and the standard deviation is denoted byσ.

The excesskurtosis is given by

${\displaystyle {\gamma }_2=\frac{-6{\mathrm{\Gamma }}_1^4+12{\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_2-3{\mathrm{\Gamma }}_2^2-4{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3+{\mathrm{\Gamma }}_4}{[{\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1^2{]}^2}}$

where${\displaystyle {\mathrm{\Gamma }}_i=\mathrm{\Gamma }(1+i/k)}$. The kurtosis excess may also be written as:

${\displaystyle {\gamma }_2=\frac{{\lambda }^4\mathrm{\Gamma }(1+\frac{4}{k})-4{\gamma }_1{\sigma }^3\mu -6{\mu }^2{\sigma }^2-{\mu }^4}{{\sigma }^4}-3.}$

A variety of expressions are available for the moment generating function ofX itself. As apower series, since the raw moments are already known, one has

${\displaystyle \mathrm{E}\left[e^{tX}\right]=\sum _{n=0}^{\mathrm{\infty }}\frac{t^n{\lambda }^n}{n!}\mathrm{\Gamma }\left(1+\frac{n}{k}\right).}$

Alternatively, one can attempt to deal directly with the integral

${\displaystyle \mathrm{E}\left[e^{tX}\right]={\int }_0^{\mathrm{\infty }}{e}^{tx}\frac{k}{\lambda }{\left(\frac{x}{\lambda }\right)}^{k-1}{e}^{-(x/\lambda {)}^k}dx.}$

If the parameterk is assumed to be a rational number, expressed ask =p/q wherep andq are integers, then this integral can be evaluated analytically.^[a] Witht replaced by −t, one finds

${\displaystyle \mathrm{E}\left[e^{-tX}\right]=\frac{1}{{\lambda }^k{t}^k}\frac{p^k\sqrt{q/p}}{(\sqrt{2\pi }{)}^{q+p-2}}{G}_{p,q}^{q,p}\left(\begin{array}{c}\frac{1-k}{p},\frac{2-k}{p},\dots ,\frac{p-k}{p}\\ \frac{0}{q},\frac{1}{q},\dots ,\frac{q-1}{q}\end{array}|\frac{p^p}{{\left(q{\lambda }^k{t}^k\right)}^q}\right)}$

whereG is theMeijer G-function.

Thecharacteristic function has also been obtained byMuraleedharan et al. (2007). Thecharacteristic function andmoment generating function of 3-parameter Weibull distribution have also been derived byMuraleedharan & Soares (2014) by a direct approach.

Let${\displaystyle X_1,X_2,\dots ,X_n}$ be independent and identically distributed Weibull random variables with scale parameter${\displaystyle \lambda }$ and shape parameter${\displaystyle k}$. If the minimum of these${\displaystyle n}$ random variables is${\displaystyle Z=min(X_1,X_2,\dots ,X_n)}$, then the cumulative probability distribution of${\displaystyle Z}$ is given by

${\displaystyle F(z)=1-e^{-n(z/\lambda {)}^k}.}$

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

Fix some${\displaystyle \alpha >0}$. Let${\displaystyle ({\pi }_1,...,{\pi }_n)}$ be nonnegative, and not all zero, and let${\displaystyle g_1,...,g_n}$ be independent samples of${\displaystyle \text{Weibull}(1,{\alpha }^{-1})}$, then^[13]

• ${\displaystyle \arg \underset{i}{min}(g_i{\pi }_i^{-\alpha })\sim \text{Categorical}{\left(\frac{{\pi }_j}{\sum _i{\pi }_i}\right)}_j}$
• ${\displaystyle \underset{i}{min}(g_i{\pi }_i^{-\alpha })\sim \text{Weibull}\left({\left(\sum _i{\pi }_i\right)}^{-\alpha },{\alpha }^{-1}\right)}$.

Theinformation entropy is given by^[14]

${\displaystyle H(\lambda ,k)=\gamma \left(1-\frac{1}{k}\right)+\ln \left(\frac{\lambda }{k}\right)+1}$

where${\displaystyle \gamma }$ is theEuler–Mascheroni constant. The Weibull distribution is themaximum entropy distribution for a non-negative real random variate with a fixedexpected value ofx^k equal toλ^k and a fixed expected value of ln(x^k) equal to ln(λ^k) − ${\displaystyle \gamma }$.

TheKullback–Leibler divergence between two Weibull distributions is given by^[15]

${\displaystyle D_{\text{KL}}({\mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}}_1\parallel {\mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}}_2)=\log \frac{k_1}{{\lambda }_1^{k_1}}-\log \frac{k_2}{{\lambda }_2^{k_2}}+(k_1-k_2)\left[\log {\lambda }_1-\frac{\gamma }{k_1}\right]+{\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}^{k_2}\mathrm{\Gamma }\left(\frac{k_2}{k_1}+1\right)-1}$

The fit of a Weibull distribution to data can be visually assessed using a Weibull plot.^[16] The Weibull plot is a plot of theempirical cumulative distribution function${\displaystyle \hat{F}(x)}$ of data on special axes in a type ofQ–Q plot. The axes are${\displaystyle \ln (-\ln (1-\hat{F}(x)))}$ versus${\displaystyle \ln (x)}$. The reason for this change of variables is the cumulative distribution function can be linearized:

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

${\displaystyle \hat{F}=\frac{i-0.3}{n+0.4}}$,

where${\displaystyle i}$ is the rank of the data point and${\displaystyle n}$ is the number of data points.^[17][18] Another common estimator^[19] is

${\displaystyle \hat{F}=\frac{i-0.5}{n}}$.

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${\displaystyle k}$ and the scale parameter${\displaystyle \lambda }$ can also be inferred.

Thecoefficient of variation of Weibull distribution depends only on the shape parameter:^[20]

${\displaystyle C{V}^2=\frac{{\sigma }^2}{{\mu }^2}=\frac{\mathrm{\Gamma }\left(1+\frac{2}{k}\right)-{\left(\mathrm{\Gamma }\left(1+\frac{1}{k}\right)\right)}^2}{{\left(\mathrm{\Gamma }\left(1+\frac{1}{k}\right)\right)}^2}.}$

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

${\displaystyle \frac{\mathrm{\Gamma }\left(1+\frac{2}{k}\right)-{\left(\mathrm{\Gamma }\left(1+\frac{1}{k}\right)\right)}^2}{{\left(\mathrm{\Gamma }\left(1+\frac{1}{k}\right)\right)}^2}=\frac{s^2}{{\overline{x}}^2}.}$

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

${\displaystyle \hat{\lambda }=\frac{\overline{x}}{\mathrm{\Gamma }\left(1+\frac{1}{\hat{k}}\right)}.}$

Themaximum likelihood estimator for the${\displaystyle \lambda }$ parameter given${\displaystyle k}$ is^[20]

${\displaystyle \hat{\lambda }={\left(\frac{1}{n}\sum _{i=1}^n{x}_i^k\right)}^{\frac{1}{k}}}$

The maximum likelihood estimator for${\displaystyle k}$ is the solution fork of the following equation^[21]

${\displaystyle 0=\frac{\sum _{i=1}^n{x}_i^k\ln x_i}{\sum _{i=1}^n{x}_i^k}-\frac{1}{k}-\frac{1}{n}\sum _{i=1}^n\ln x_i}$

This equation defines${\displaystyle \hat{k}}$ only implicitly, one must generally solve for${\displaystyle k}$ by numerical means.

When${\displaystyle x_1>x_2>\cdots >x_N}$ are the${\displaystyle N}$ largest observed samples from a dataset of more than${\displaystyle N}$ samples, then the maximum likelihood estimator for the${\displaystyle \lambda }$ parameter given${\displaystyle k}$ is^[21]

${\displaystyle {\hat{\lambda }}^k=\frac{1}{N}\sum _{i=1}^N(x_i^k-x_N^k)}$

Also given that condition, the maximum likelihood estimator for${\displaystyle k}$ is^{[citation needed]}

${\displaystyle 0=\frac{\sum _{i=1}^N(x_i^k\ln x_i-x_N^k\ln x_N)}{\sum _{i=1}^N(x_i^k-x_N^k)}-\frac{1}{N}\sum _{i=1}^N\ln x_i}$

Again, this being an implicit function, one must generally solve for${\displaystyle k}$ by numerical means.

The Weibull distribution is used^{[citation needed]}

• Insurvival analysis
• Inreliability engineering andfailure analysis
• Inelectrical engineering to represent overvoltage occurring in an electrical system
• Inindustrial engineering to representmanufacturing anddelivery times
• Inextreme value theory
• Inweather forecasting and thewind power industry to describewind speed distributions, as the natural distribution often matches the Weibull shape^[24]
• In communications systems engineering
  • Inradar systems to model the dispersion of the received signals level produced by some types of clutters
  • To modelfading channels inwireless communications, as theWeibull fading model seems to exhibit good fit to experimental fadingchannel measurements
• Ininformation retrieval to model dwell times on web pages.^[25]
• Ingeneral insurance to model the size ofreinsurance claims, and the cumulative development ofasbestosis losses
• In forecasting technological change (also known as the Sharif-Islam model)^[26]
• Inhydrology the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.
• Indecline curve analysis to model oil production rate curve of shale oil wells.^[23]
• In describing the size ofparticles generated by grinding,milling andcrushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin–Rammler distribution.^[27] In this context it predicts fewer fine particles than thelog-normal distribution and it is generally most accurate for narrow particle size distributions.^[28] The interpretation of the cumulative distribution function is that${\displaystyle F(x;k,\lambda )}$ is themass fraction of particles with diameter smaller than${\displaystyle x}$, where${\displaystyle \lambda }$ is the mean particle size and${\displaystyle k}$ 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${\displaystyle x}$ from a given particle is given by a Weibull distribution with${\displaystyle k=3}$ and${\displaystyle \rho =1/{\lambda }^3}$ equal to the density of the particles.^[29]
• In calculating the rate of radiation-inducedsingle event effects onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured devicecross section probability data to a particlelinear energy transfer spectrum.^[30] 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^{[citation needed]} and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.^[31]

• If${\displaystyle W\sim \mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}(\lambda ,k)}$, then the variable${\displaystyle G=\log W}$ is Gumbel (minimum) distributed with location parameter${\displaystyle \mu =\log \lambda }$ and scale parameter${\displaystyle \beta =1/k}$. That is,${\displaystyle G\sim {\mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}}_{min}(\log \lambda ,1/k)}$.
• A Weibull distribution is ageneralized gamma distribution with both shape parameters equal tok.
• The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter.^[12] It has theprobability density function

  ${\displaystyle f(x;k,\lambda ,\theta )=\frac{k}{\lambda }{\left(\frac{x-\theta }{\lambda }\right)}^{k-1}{e}^{-{\left(\frac{x-\theta }{\lambda }\right)}^k}}$

  for${\displaystyle x\ge \theta }$ and${\displaystyle f(x;k,\lambda ,\theta )=0}$ for, where${\displaystyle k>0}$ is theshape parameter,${\displaystyle \lambda >0}$ is thescale parameter and${\displaystyle \theta }$ is thelocation parameter of the distribution.${\displaystyle \theta }$ value sets an initial failure-free time before the regular Weibull process begins. When${\displaystyle \theta =0}$, this reduces to the 2-parameter distribution.
• The Weibull distribution can be characterized as the distribution of a random variable${\displaystyle W}$ such that the random variable

  ${\displaystyle X={\left(\frac{W}{\lambda }\right)}^k}$

  is the standardexponential distribution with intensity 1.^[12]
• This implies that the Weibull distribution can also be characterized in terms of auniform distribution: if${\displaystyle U}$ is uniformly distributed on${\displaystyle (0,1)}$, then the random variable${\displaystyle W=\lambda (-\ln (U){)}^{1/k}}$ is Weibull distributed with parameters${\displaystyle k}$ and${\displaystyle \lambda }$. Note that${\displaystyle -\ln (U)}$ here is equivalent to${\displaystyle X}$ 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${\displaystyle 1/\lambda }$ when${\displaystyle k=1}$ and aRayleigh distribution of mode${\displaystyle \sigma =\lambda /\sqrt{2}}$ when${\displaystyle k=2}$.
• The Weibull distribution (usually sufficient inreliability engineering) is a special case of the three parameterexponentiated Weibull distribution where the additional exponent equals 1. The exponentiated Weibull distribution accommodatesunimodal,bathtub shaped^[32] andmonotonefailure rates.
• The Weibull distribution is a special case of thegeneralized extreme value distribution. It was in this connection that the distribution was first identified byMaurice Fréchet in 1927.^[33] The closely relatedFréchet distribution, named for this work, has the probability density function

  ${\displaystyle f_{\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{t}}(x;k,\lambda )=\frac{k}{\lambda }{\left(\frac{x}{\lambda }\right)}^{-1-k}{e}^{-(x/\lambda {)}^{-k}}=f_{\mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}}(x;-k,\lambda ).}$

• The distribution of a random variable that is defined as the minimum of several random variables, each having a different Weibull distribution, is apoly-Weibull distribution.
• The Weibull distribution was first applied byRosin & Rammler (1933) to describe particle size distributions. It is widely used inmineral processing to describeparticle size distributions incomminution processes. In this context the cumulative distribution is given by

  

  where
  • ${\displaystyle x}$ is the particle size
  • ${\displaystyle P_{80}}$ is the 80th percentile of the particle size distribution
  • ${\displaystyle m}$ is a parameter describing the spread of the distribution
• Because of its availability inspreadsheets, it is also used where the underlying behavior is actually better modeled by anErlang distribution.^[34]
• If${\displaystyle X\sim \mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}(\lambda ,\frac{1}{2})}$ then${\displaystyle \sqrt{X}\sim \mathrm{E}\mathrm{x}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}(\frac{1}{\sqrt{\lambda }})}$ (Exponential distribution)
• For the same values of k, theGamma distribution takes on similar shapes, but the Weibull distribution is moreplatykurtic.

• From the viewpoint of theStable count distribution,${\displaystyle k}$ 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 aLaplace distribution${\displaystyle F(x;1,\lambda )}$ or aRayleigh distribution${\displaystyle F(x;2,\lambda )}$:

  

  where${\displaystyle {\mathfrak{N}}_k(\nu )}$ is theStable count distribution and${\displaystyle V_k(s)}$ is theStable vol distribution.

• Discrete Weibull distribution
• Fisher–Tippett–Gnedenko theorem
• Logistic distribution
• Rosin–Rammler distribution for particle size analysis
• Rayleigh distribution
• Stable count distribution

• "Weibull distribution",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Mathpages – Weibull analysis
• The Weibull Distribution
• Reliability Analysis with Weibull
• Interactive graphic:Univariate Distribution Relationships
• Online Weibull Probability Plotting

• Continuous distributions
• Survival analysis
• Exponential family distributions
• Extreme value data

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