Weibull (2-parameter)
Probability density function
Probability distribution function
Cumulative distribution function
Cumulative distribution function
Parameters scale
Excess kurtosis (see text)
Kullback–Leibler divergence see below

In probability theory and statistics, the Weibull distribution /ˈwbʊl/ 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,[1] although it was first identified by René Maurice Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.


Standard parameterization

The probability density function of a Weibull random variable is[2][3]

where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2 and [4]).

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:[5]

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.[7][8] The shape parameter k is the same as above, while the scale parameter is . In this case, for x ≥ 0, the probability density function is

the cumulative distribution function is

the hazard function is

and the mean is

Second alternative

A second alternative parameterization can also be found.[9][10] The shape parameter k is the same as in the standard case, while the scale parameter λ is replaced with a rate parameter β = 1/λ. Then, for x ≥ 0, the probability density function is

the cumulative distribution function is

and the hazard function is

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.


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 distribution of type III.

Cumulative distribution function

The cumulative distribution function for the Weibull distribution is

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

If x = λ then F(x; k; λ) = 1 − e−1 ≈ 0.632 for all values of k. Vice versa: at F(x; k; λ) = 0.632 the value of x ≈ λ.

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

for 0 ≤ p < 1.

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

The Mean time between failures MTBF is


The moment generating function of the logarithm of a Weibull distributed random variable is given by[11]

where Γ is the gamma function. Similarly, the characteristic function of log X is given by

In particular, the nth raw moment of X is given by

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


The skewness is given by

where , which may also be written as

where the mean is denoted by μ and the standard deviation is denoted by σ.

The excess kurtosis is given by

where . The kurtosis excess may also be written as:

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

Alternatively, one can attempt to deal directly with the integral

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.[12] With t replaced by −t, one finds

where G is the Meijer G-function.

The characteristic function has also been obtained by Muraleedharan et al. (2007). The characteristic function and moment generating function of 3-parameter Weibull distribution have also been derived by Muraleedharan & Soares (2014) by a direct approach.


Let be independent and identically distributed Weibull random variables with scale parameter and shape parameter . If the minimum of these random variables is , then the cumulative probability distribution of given by

That is, will also be Weibull distributed with scale parameter and with shape parameter .

Reparametrization tricks

Fix some . Let be nonnegative, and not all zero, and let be independent samples of , then[13]

Shannon entropy

The information entropy is given by

where 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 xk equal to λk and a fixed expected value of ln(xk) equal to ln(λk) − .

Kullback–Leibler divergence

The Kullback–Leibler divergence between two Weibulll distributions is given by[14]

Parameter estimation

Ordinary least square using Weibull plot

Weibull plot

The fit of a Weibull distribution to data can be visually assessed using a Weibull plot.[15] The Weibull plot is a plot of the empirical cumulative distribution function of data on special axes in a type of Q–Q plot. The axes are versus . 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 where is the rank of the data point and is the number of data points.[16]

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 and the scale parameter can also be inferred.

Method of moments

The coefficient of variation of Weibull distribution depends only on the shape parameter:[17]

Equating the sample quantities to , the moment estimate of the shape parameter can be read off either from a look up table or a graph of versus . A more accurate estimate of can be found using a root finding algorithm to solve

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

Maximum likelihood

The maximum likelihood estimator for the parameter given is[17]

The maximum likelihood estimator for is the solution for k of the following equation[18]

This equation defines only implicitly, one must generally solve for by numerical means.

When are the largest observed samples from a dataset of more than samples, then the maximum likelihood estimator for the parameter given is[18]

Also given that condition, the maximum likelihood estimator for is[citation needed]

Again, this being an implicit function, one must generally solve for by numerical means.


The Weibull distribution is used[citation needed]

Fitted cumulative Weibull distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting[19]
Fitted curves for oil production time series data [20]

Related distributions

See also


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  2. ^ Papoulis, Athanasios Papoulis; Pillai, S. Unnikrishna (2002). Probability, Random Variables, and Stochastic Processes (4th ed.). Boston: McGraw-Hill. ISBN 0-07-366011-6.
  3. ^ Kizilersu, Ayse; Kreer, Markus; Thomas, Anthony W. (2018). "The Weibull distribution". Significance. 15 (2): 10–11. doi:10.1111/j.1740-9713.2018.01123.x.
  4. ^ "Rayleigh Distribution – MATLAB & Simulink – MathWorks Australia". www.mathworks.com.au.
  5. ^ Jiang, R.; Murthy, D.N.P. (2011). "A study of Weibull shape parameter: Properties and significance". Reliability Engineering & System Safety. 96 (12): 1619–26. doi:10.1016/j.ress.2011.09.003.
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  10. ^ Therneau, T. (2020). "A Package for Survival Analysis in R." R package version 3.1.
  11. ^ a b c Johnson, Kotz & Balakrishnan 1994
  12. ^ See (Cheng, Tellambura & Beaulieu 2004) for the case when k is an integer, and (Sagias & Karagiannidis 2005) for the rational case.
  13. ^ Balog, Matej; Tripuraneni, Nilesh; Ghahramani, Zoubin; Weller, Adrian (2017-07-17). "Lost Relatives of the Gumbel Trick". International Conference on Machine Learning. PMLR: 371–379.
  14. ^ Bauckhage, Christian (2013). "Computing the Kullback-Leibler Divergence between two Weibull Distributions". arXiv:1310.3713 [cs.IT].
  15. ^ " Weibull Plot". www.itl.nist.gov.
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  19. ^ "CumFreq, Distribution fitting of probability, free software, cumulative frequency".
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  21. ^ "Wind Speed Distribution Weibull – REUK.co.uk". www.reuk.co.uk.
  22. ^ Liu, Chao; White, Ryen W.; Dumais, Susan (2010-07-19). Understanding web browsing behaviors through Weibull analysis of dwell time. ACM. pp. 379–386. doi:10.1145/1835449.1835513. ISBN 9781450301534. S2CID 12186028.
  23. ^ Sharif, M.Nawaz; Islam, M.Nazrul (1980). "The Weibull distribution as a general model for forecasting technological change". Technological Forecasting and Social Change. 18 (3): 247–56. doi:10.1016/0040-1625(80)90026-8.
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