Weibull distribution

Weibull (2-Parameter)

Probability density function
Cumulative distribution function
Parameters	$\lambda >0\,$ scale (real) $k>0\,$ shape (real)
Support	$x\in [0;+\infty )\,$
PDF	$f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}&x\geq 0\\0&x<0\end{cases}}$
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	$\gamma \left(1\!-\!{\frac {1}{k}}\right)+\ln \left({\frac {\lambda }{k}}\right)+1$
MGF	$\sum _{n=0}^{\infty }{\frac {t^{n}\lambda ^{n}}{n!}}\Gamma \left(1+{\frac {n}{k}}\right),\ k\geq 1$
CF	$\sum _{n=0}^{\infty }{\frac {(it)^{n}\lambda ^{n}}{n!}}\Gamma \left(1+{\frac {n}{k}}\right)$

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:

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

where $k>0$ is the shape parameter and $\lambda >0$ 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 ($k=2$).

It gives the distribution of failures, where the failure rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so can be interpreted directly as follows:

• A value of k<1 indicates that the failure rate decreases over time. 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.
• A value of k=1 indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure.
• A value of 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.

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

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

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 $\Gamma _{i}=\Gamma (1+i/k)$. 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. With t replaced by −t, one finds

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

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 $\gamma$ is the Euler–Mascheroni constant.

Related distributions

The spatially shifted Weibull distribution contains an additional parameter, and is also often found in the literature. 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, $\lambda >0$ is the scale parameter and $\theta$ 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. 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=λ.

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 (usually sufficient in reliability engineering) 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* and monotone failure rates.

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

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

The Weibull distribution can also be generalized to the 3 parameter exponentiated Weibull distribution. This models the situation when the failure rate of a system is due to a combination of factors, and may increase for some times and decrease for other times (see bathtub curve).

Uses

The Weibull distribution is used

• In survival analysis
• In reliability engineering and failure analysis
• In industrial engineering to represent manufacturing and delivery times
• In extreme value theory
• In weather forecasting
  • To describe wind speed distributions, as the natural distribution often matches the Weibull shape
• In communications systems engineering
  • 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)

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

See also

• Exponentiated Weibull distribution
• Weibull modulus

Bibliography

• Fréchet, Maurice (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la Société Polonaise de Mathematique, Cracovie, 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, ISBN 978-0-471-58495-7, MR1299979
• Muraleedharan, G.; Rao, A.G.; Kurup, P.G.; Nair, N. Unnikrishnan; Sinha, Mourani (2007), Coastal Engineering, vol. 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, ISSN 0018-9448, MR2237527
• 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. {{cite web}}: |chapter= ignored (help)
• Nelson, Jr, Ralph (2008-02-05). "Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution". Retrieved 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.