Weibull distribution: Definition from Answers.com

Weibull (2-Parameter)
Probability density function
Cumulative distribution function
Parameters	$\lambda>0\,$ scale (real) $k>0\,$ shape (real)
Support	$x \in [0; +\infty)\,$
Probability density function (pdf)	$f(x)=\begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0\end{cases}$
Cumulative distribution function (cdf)	$1- e^{-(x/\lambda)^k}$
Mean	$\lambda \Gamma\left(1+\frac{1}{k}\right)\,$
Median	$\lambda(\ln(2))^{1/k}\,$
Mode	$\lambda \left(\frac{k-1}{k} \right)^{\frac{1}{k}}\,$ if $k > 1$
Variance	$\lambda^2\Gamma\left(1+\frac{2}{k}\right) - \mu^2\,$
Skewness	$\frac{\Gamma(1+\frac{3}{k})\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}$
Excess kurtosis	(see text)
Entropy	$\gamma\left(1\!-\!\frac{1}{k}\right)+\ln\left(\frac{\lambda}{k}\right)+1$
Moment-generating function (mgf)	$\sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma\left(1+\frac{n}{k}\right)$
Characteristic function	$\sum_{n=0}^\infty \frac{(it)^n\lambda^n}{n!}\Gamma\left(1+\frac{n}{k}\right)$

In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Waloddi Weibull who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe the size distribution of particles. The probability density function of a Weibull random variable x is^[1]:

$f(x;\lambda,k) = \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0\end{cases}$

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 ( $k = 2$ ).

The Weibull distribution is often used in the field of life data analysis due to its flexibility—it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then k < 1. If the failure rate is constant over time, then k = 1. If the failure rate increases over time, then k > 1.

An understanding of the failure rate may provide insight as to what is causing the failures:

A decreasing failure rate would suggest "infant mortality". That is, defective items fail early and the failure rate decreases over time as they fall out of the population.

A constant failure rate suggests that items are failing from random events.

An increasing failure rate suggests "wear out" - parts are more likely to fail as time goes on.

1 Properties
2 Related distributions
3 Uses
4 References
5 Bibliography
6 External links

Properties

The cumulative distribution function for the Weibull distribution is

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

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

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

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

Moments

The moment generating function of the logarithm of a Weibull distributed random variable is given by^[2]

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

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

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

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

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

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

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

and

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

The skewness is given by:

$\gamma_1=\frac{\Gamma\left(1+\frac{3}{k}\right)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}.$

The excess kurtosis is given by:

$\gamma_2=\frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2 -4\Gamma_1\Gamma_3+\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2}$

where $Γ i = Γ(1 + i / k)$ . The kurtosis excess may also be written as :

$\gamma_{2}=\frac{\lambda^4\Gamma(1+\frac{4}{k})-4\gamma_{1}\sigma^3\mu-6\mu^2\sigma^2-\mu^4}{\sigma^4}$

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

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

Alternatively, one can attempt to deal directly with the integral

$E\left[e^{tX}\right] = \int_0^\infty e^{tx} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}\,dx.$

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.^[3] With t replaced by −t, one finds

$E\left[e^{-tX}\right]= \frac1{ \lambda^k\, t^k} \, \frac{ p^k \, \sqrt{q/p}}{ ( \sqrt{2 \pi} )^{q+p-2}} \, G^{q \; p}_{p \; q} \left[ \left. \frac{ p^p }{\left( q \, \lambda^k \, t^k \right)^q} \right| ^{ (1- k) / p, \, (2 - k) / p, \ldots, (p- k) / p }_{ 0/q, \, 1 / q, \ldots, (q-1) / q} \right]$

where G is the Meijer G-function.

The characteristic function has also been obtained by Muraleedharan et al. (2007).

Information entropy

The information entropy is given by

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

where $γ$ is the Euler–Mascheroni constant.

Related distributions

The spatially shifted Weibull distribution contains an additional parameter, and is also often found in the literature.^[4] It has the probability density function

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

for $x \geq \theta$ and f(x; k, λ, θ) = 0 for x < θ, where $k > 0$ is the shape parameter, $λ > 0$ is the scale parameter and $θ$ is the location parameter of the distribution. When θ=0, this reduces to the 2-parameter distribution.

The Weibull distribution can be characterized as the distribution of a random variable X such that the random variable

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

is the standard exponential distribution with intensity 1.^[5] The Weibull distribution interpolates between the exponential distribution with intensity 1/λ when k = 1 and a Rayleigh distribution of mode $\sigma = \lambda\sqrt{2}$ when k = 2.

The density function of the Weibull distribution changes character radically as k varies between 0 and 3, particularly in terms of its behaviour near x=0. For k < 1 the density approaches ∞ as x nears zero and the density is J-shaped. For k = 1 the density has a finite positive value at x=0. For 1<k<2 the density is zero nears zero,has an infinite slope at x=0 and is unimodal. For k=2 the density has a finite positive slope at x=0. For k>2 the density is zero and has a zero slope at x=0 and the density is unimodal. As k goes to infinity, the Weibull distribution converges to a Dirac delta distribution centred at x=1.

The Weibull distribution can also be characterized in terms of a uniform distribution: if X is uniformly distributed on (0,1), then the random variable $\lambda(-\ln(X))^{1/k}\,$ Weibull distributed with parameters k and λ. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.

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. The closely related Fréchet distribution, named for this work, has the probability density function

$f_{\rm{Frechet}}(x;k,\lambda)=\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{-1-k} e^{-(x/\lambda)^{-k}} = -f_{\rm{Weibull}}(x;-k,\lambda).$

Uses

The Weibull distribution is used

In survival analysis
To represent manufacturing and delivery times in industrial engineering
In extreme value theory
In weather forecasting
In reliability engineering and failure analysis
In radar systems to model the dispersion of the received signals level produced by some types of clutters
To model fading channels in wireless communications, as the Weibull fading model seems to exhibit good fit to experimental fading channel measurements
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)
To describe wind speed distributions, as the natural distribution often matches the Weibull shape

The Weibull distribution may be used in place of the normal distribution because a Weibull variate can be generated through inversion. Normal variates are typically generated using the more complicated Box-Muller method, which requires two uniform random variates.

The 2-Parameter Weibull distribution is used to describe the particle size distribution of particles generated by grinding, milling and crushing operations. The Rosin-Rammler distribution predicts fewer fine particles than the Log-normal distribution. It is generally most accurate for narrow PSDs.

Using the cumulative distribution function:

F(x; k; λ) is the mass fraction of particles with diameter < x
λ is the mean particle size
k is a measure of particle size spread

References

^ Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes, 4th Edition
^ Johnson, Kotz & Balakrishnan 1994
^ See (Cheng, Tellambura & Beaulieu 2004) for the case when k is an integer, and (Sagias & Karagiannidis 2005) for the rational case.
^ Johnson, Kotz & Balakrishnan 1994
^ Johnson, Kotz & Balakrishnan 1994

Bibliography

Fréchet, Maurice (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la Société Polonaise de Mathematique, Crocovie 6: 93-116 .
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, MR 1299979, ISBN 978-0-471-58495-7
Muraleedharan, G.; Rao, A.G.; Kurup, P.G.; Nair, N. Unnikrishnan; Sinha, Mourani (2007), Coastal Engineering, 54, pp. 630-638, doi:10.1016/j.coastaleng.2007.05.001
Rosin, P.; Rammler, E. (1933), "The Laws Governing the Fineness of Powdered Coal", Journal of the Institute of Fuel 7: 29 - 36 .
Sagias, Nikos C.; Karagiannidis, George K. (2005), "Gaussian class multivariate Weibull distributions: theory and applications in fading channels", Institute of Electrical and Electronics Engineers. Transactions on Information Theory 51 (10): 3608–3619, doi:10.1109/TIT.2005.855598, MR 2237527, ISSN 0018-9448
Weibull, W. (1951), "A statistical distribution function of wide applicability", J. Appl. Mech.-Trans. ASME 18 (3): 293-297 .
"Engineering statistics handbook". National Institute of Standards and Technology. 2008. http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm.
Nelson, Jr, Ralph (2008-02-05). "Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution". http://www.erpt.org/014Q/nelsa-06.htm. Retrieved on 2008-02-05.

External links

The Weibull distribution (with examples, properties, and calculators).
The Weibull plot.
Weibull plotting paper.
Mathpages - Weibull Analysis
Using Excel for Weibull Analysis
This article from the Quality Digest describes how to use MS Excel to analyse lifetest data with the Weibull statistical distribution. Although Excel doesn't have an inverse Weibull function, this article shows how to use Excel to solve for critical values.
The SOCR Resource provides interactive interface to Weibull distribution.

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin-Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · stable · Tweedie

v • d • e Common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

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	Ernst Hjalmar Wallodi Weibull
	Weibull

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