# Weibull distribution

Parameters Probability density function Cumulative distribution function $\lambda>0\,$ scale (real) $k>0\,$ shape (real) $x \in [0; +\infty)\,$ $f(x)=\begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0\end{cases}$ $1- e^{-(x/\lambda)^k}$ $\lambda \Gamma\left(1+\frac{1}{k}\right)\,$ $\lambda(\ln(2))^{1/k}\,$ $\lambda \left(\frac{k-1}{k} \right)^{\frac{1}{k}}\,$ if k > 1 $\lambda^2\Gamma\left(1+\frac{2}{k}\right) - \mu^2\,$ $\frac{\Gamma(1+\frac{3}{k})\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}$ (see text) $\gamma\left(1\!-\!\frac{1}{k}\right)+\ln\left(\frac{\lambda}{k}\right)+1$ see Weibull fading see [[Muraleedharan.et.al.,(2007)Coastal Engineering 54 (8), 630-638. http://dx.doi.org/10.1016/j.coastaleng.2007.05.001 see Reply to Saralees Nadarajah,(2008)Coastal Engineering 55 (2), 191-193 http://dx.doi.org/10.1016/j.coastaleng.2007.07.006]]

In probability theory and statistics, the Weibull distribution[1] (named after Waloddi Weibull) is a continuous probability distribution. It is often called the Rosin–Rammler distribution when used to describe the size distribution of particles. The distribution was introduced by P. Rosin and E. Rammler in 1933.[2][3] The probability density function of a Weibull random variable x is[4]:

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

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 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.

Under certain parameterizations, the Weibull distribution reduces to several other familiar distributions:

## Properties

Characteristic Function of Weibull Distribution

The Characteristic Function (φ(t)) of Weibull distribution is given by Muraleedharan; et al (2007), Coastal Engineering, 54, pp. 630-638 . It can generate all the moments of Weibull distribution and hence characteristic functions are also known as moment generating functions(mgf). It also gives all the moments of Rayleigh and exponential distributions for b (shape parameter)equal to 2 and 1 respectively. Also it gives the characteristic and moment generating functions of Rayleigh and exponential distributions for b equal to 2 and 1 respectively. It satisfied all the conditions for a function to be a characteristic function (See Reply to Saralees Nadarajah (2008),Coastal Engineering 55(2),191-193).

Also by Taylor's Theorem for complex functions,the characteristic function of any random variable, X, with finite mean μ can be written as

φ(t)=1+itμ+0(t); t→0

This is also true for the characteristic function of Weibull,Rayleigh (b=2) and exponential (b=1) distributions as

μ= aГ(1+1/b)

Hence the characteristic function given by Muraleedharan et.al(2007) is correct. ______________________________________________________________________________________________

There is an abrupt change in the value of the density function at 0 when k takes on values around 1. It is because,[5]

• for any $k < 1, f(x) \rightarrow \infty$ as $x \rightarrow 0$
• for $k=1, f(0)=\frac{1}{\lambda}$, and
• for any $k > 1, f(x) \rightarrow 0$ as $x \rightarrow 0$

The nth raw moment is given by:

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

where Γ is the Gamma function. 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 Γi = Γ(1 + i / k). The kurtosis excess may also be written as :

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

A generalized, 3-parameter Weibull distribution 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, 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 cumulative distribution function for the 2-parameter Weibull is

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

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

The cumulative distribution function for the 3-parameter Weibull is

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

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

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

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

### 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.

## Generating Weibull-distributed random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

$X=\lambda (-\ln(U))^{1/k}\,$

has a Weibull distribution with parameters k and λ. This follows from the form of the cumulative distribution function. Note that if you are generating random numbers belonging to (0,1), exclude zero values to avoid the natural log of zero.

## Related distributions

• $X \sim \mathrm{Exponential}(\lambda)$ is an exponential distribution if X˜Weibull(k = 1,λ − 1).
• $X \sim \mathrm{Rayleigh}(\beta)$ is a Rayleigh distribution if $X \sim \mathrm{Weibull}(k = 2, \sqrt{2} \beta)$.
• $\lambda(-\ln(X))^{1/k}\,$ is a Weibull distribution if $X \sim \mathrm{Uniform}(0,1)$.
• Inverse Weibull distribution with p.d.f. $f(x;k,\lambda)=(k/\lambda) (\lambda/x)^{(k+1)} e^{-(\lambda/x)^k}$

## Uses

The Weibull distribution is used

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

1. ^ Weibull, W. (1951) "A statistical distribution function of wide applicability" J. Appl. Mech.-Trans. ASME 18(3), 293-297
2. ^ http://www.zarm.uni-bremen.de/gamm98/num_abs/a912.html
3. ^ Rosin, P., and Rammler, E. (1933) "The Laws Governing the Fineness of Powdered Coal", Journal of the Institute of Fuel, vol. 7, pp. 29 - 36.
4. ^ Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes, 4th Edition
5. ^ http://www.weibull.com/LifeDataWeb/characteristics_of_the_weibull_distribution.htm