Weibull distribution

In probability theory and statistics, the Weibull distribution /ˈwaɪbʊl/ 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.

Parameters

$\lambda \in (0,+\infty )\,$ scale
$k\in (0,+\infty )\,$ shape

$x\in [0,+\infty )\,$

$f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0.\end{cases}}$

$F(x)={\begin{cases}1-e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0.\end{cases}}$

$Q(p)=\lambda (-\ln(1-p))^{\frac {1}{k}}$

$\lambda \,\Gamma (1+1/k)\,$

$\lambda (\ln 2)^{1/k}\,$

${\begin{cases}\lambda \left({\frac {k-1}{k}}\right)^{1/k}\,,&k>1,\\0,&k\leq 1.\end{cases}}$

$\lambda ^{2}\left[\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}\right]\,$

${\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}$

(see text)

$\gamma (1-1/k)+\ln(\lambda /k)+1\,$

$\sum _{n=0}^{\infty }{\frac {t^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k),\ k\geq 1$

$\sum _{n=0}^{\infty }{\frac {(it)^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k)$

see below

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.

Definition[edit]

Standard parameterization[edit]

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

Properties[edit]

Density function[edit]

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[edit]

The cumulative distribution function for the Weibull distribution is

In

survival analysis

In and failure analysis

reliability engineering

In to represent overvoltage occurring in an electrical system

electrical engineering

In to represent manufacturing and delivery times

industrial engineering

In

extreme value theory

In and the wind power industry to describe wind speed distributions, as the natural distribution often matches the Weibull shape^[22]

weather forecasting

radar

In to model dwell times on web pages.^[23]

information retrieval

In to model the size of reinsurance claims, and the cumulative development of asbestosis losses

general insurance

In forecasting technological change (also known as the Sharif-Islam model)

[24]

In the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.

hydrology

In to model oil production rate curve of shale oil wells.^[21]

decline curve analysis

In describing the size of generated by grinding, milling and crushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin–Rammler distribution.^[25] In this context it predicts fewer fine particles than the log-normal distribution and it is generally most accurate for narrow particle size distributions.^[26] The interpretation of the cumulative distribution function is that $F(x;k,\lambda )$ is the mass fraction of particles with diameter smaller than $x$ , where $\lambda$ is the mean particle size and $k$ is a measure of the spread of particle sizes.

particles

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 $x$ from a given particle is given by a Weibull distribution with $k=3$ and $\rho =1/\lambda ^{3}$ equal to the density of the particles.

[27]

In calculating the rate of radiation-induced onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured device cross section probability data to a particle linear energy transfer spectrum.^[28] 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 and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.^[29]

single event effects

The Weibull distribution is used

If $W\sim \mathrm {Weibull} (\lambda ,k)$ , then the variable $G=\log W$ is Gumbel (minimum) distributed with location parameter $\mu =\log \lambda$ and scale parameter $\beta =1/k$ . That is, $G\sim \mathrm {Gumbel} _{\min }(\log \lambda ,1/k)$ .

A Weibull distribution is a with both shape parameters equal to k.

generalized gamma distribution

The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter. It has the probability density function
$f(x;k,\lambda ,\theta )={k \over \lambda }\left({x-\theta \over \lambda }\right)^{k-1}e^{-\left({x-\theta \over \lambda }\right)^{k}}\,$

for $x\geq \theta$ and $f(x;k,\lambda ,\theta )=0$ for $x<\theta$ , where $k>0$ is the shape parameter, $\lambda >0$ is the scale parameter and $\theta$ is the location parameter of the distribution. $\theta$ value sets an initial failure-free time before the regular Weibull process begins. When $\theta =0$ , this reduces to the 2-parameter distribution.

[11]

The Weibull distribution can be characterized as the distribution of a random variable $W$ such that the random variable
$X=\left({\frac {W}{\lambda }}\right)^{k}$

is the standard with intensity 1.^[11]

exponential distribution

This implies that the Weibull distribution can also be characterized in terms of a : if $U$ is uniformly distributed on $(0,1)$ , then the random variable $W=\lambda (-\ln(U))^{1/k}\,$ is Weibull distributed with parameters $k$ and $\lambda$ . Note that $-\ln(U)$ here is equivalent to $X$ just above. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.

uniform distribution

The Weibull distribution interpolates between the exponential distribution with intensity $1/\lambda$ when $k=1$ and a of mode $\sigma =\lambda /{\sqrt {2}}$ when $k=2$ .

Rayleigh distribution

The Weibull distribution (usually sufficient in ) is a special case of the three parameter exponentiated Weibull distribution where the additional exponent equals 1. The exponentiated Weibull distribution accommodates unimodal, bathtub shaped^[30] and monotone failure rates.

reliability engineering

The Weibull distribution is a special case of the . It was in this connection that the distribution was first identified by Maurice Fréchet in 1927.^[31] The closely related Fréchet distribution, named for this work, has the probability density function
$f_{\rm {Frechet}}(x;k,\lambda )={\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{-1-k}e^{-(x/\lambda )^{-k}}=f_{\rm {Weibull}}(x;-k,\lambda ).$

generalized extreme value distribution

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

Because of its availability in , it is also used where the underlying behavior is actually better modeled by an Erlang distribution.^[32]

spreadsheets

If $X\sim \mathrm {Weibull} (\lambda ,{\frac {1}{2}})$ then ${\sqrt {X}}\sim \mathrm {Exponential} ({\frac {1}{\sqrt {\lambda }}})$ ()

Exponential distribution

For the same values of k, the takes on similar shapes, but the Weibull distribution is more platykurtic.

Gamma distribution

Discrete Weibull distribution

Fisher–Tippett–Gnedenko theorem

Logistic distribution

for particle size analysis

Rosin–Rammler distribution

Rayleigh distribution

Stable count distribution

(1927), "Sur la loi de probabilité de l'écart maximum", Annales de la Société Polonaise de Mathématique, Cracovie, 6: 93–116.

Fréchet, Maurice

Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, 978-0-471-58495-7, MR 1299979

ISBN

; Schafer, Ray E.; Singpurwalla, Nozer D. (1974), Methods for Statistical Analysis of Reliability and Life Data, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (1st ed.), New York: John Wiley & Sons, ISBN 978-0-471-56737-0

Mann, Nancy R.

Muraleedharan, G.; Rao, A.D.; Kurup, P.G.; Nair, N. Unnikrishnan; Sinha, Mourani (2007), "Modified Weibull Distribution for Maximum and Significant Wave Height Simulation and Prediction", Coastal Engineering, 54 (8): 630–638, :10.1016/j.coastaleng.2007.05.001

doi

Rosin, P.; Rammler, E. (1933), "The Laws Governing the Fineness of Powdered Coal", Journal of the Institute of Fuel, 7: 29–36.

Sagias, N.C.; Karagiannidis, G.K. (2005). "Gaussian Class Multivariate Weibull Distributions: Theory and Applications in Fading Channels". IEEE Transactions on Information Theory. 51 (10): 3608–19. :10.1109/TIT.2005.855598. MR 2237527. S2CID 14654176.

doi

(1951), "A statistical distribution function of wide applicability" (PDF), Journal of Applied Mechanics, 18 (3): 293–297, Bibcode:1951JAM....18..293W, doi:10.1115/1.4010337.

Weibull, W.

. Engineering statistics handbook. National Institute of Standards and Technology. 2008.

"Weibull Distribution"

Nelson Jr, Ralph (2008-02-05). . Retrieved 2008-02-05.

"Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution"

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Weibull distribution"

Mathpages – Weibull analysis

The Weibull Distribution

Reliability Analysis with Weibull

Interactive graphic:

Weibull distribution

Definition[edit]

Standard parameterization[edit]

Properties[edit]

Density function[edit]

Cumulative distribution function[edit]

survival analysis

reliability engineering

electrical engineering

industrial engineering

extreme value theory

weather forecasting

radar

information retrieval

general insurance

[24]

hydrology

decline curve analysis

particles

[27]

single event effects

generalized gamma distribution

[11]

exponential distribution

uniform distribution

Rayleigh distribution

reliability engineering

generalized extreme value distribution

poly-Weibull distribution

spreadsheets

Exponential distribution

Gamma distribution

Discrete Weibull distribution

Fisher–Tippett–Gnedenko theorem

Logistic distribution

Rosin–Rammler distribution

Rayleigh distribution

Stable count distribution

Fréchet, Maurice

ISBN

Mann, Nancy R.

doi

doi

Weibull, W.

"Weibull Distribution"

"Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution"

"Weibull distribution"

Mathpages – Weibull analysis

The Weibull Distribution

Reliability Analysis with Weibull

Univariate Distribution Relationships

Online Weibull Probability Plotting