Weibull distribution

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 Probability density function Cumulative distribution function Parameters $\lambda>0\,$ scale (real)$k>0\,$ shape (real) Support $x \in [0; +\infty)\,$ pdf $(k/\lambda) (x/\lambda)^{(k-1)} e^{-(x/\lambda)^k}$ cdf $1- e^{-(x/\lambda)^k}$ Mean $\mu=\lambda \Gamma\left(1+\frac{1}{k}\right)\,$ Median $\lambda\ln(2)^{1/k}\,$ Mode Variance $\sigma^2=\lambda^2\Gamma\left(1+\frac{2}{k}\right) - \mu^2\,$ Skewness $\frac{\Gamma(1+\frac{3}{k})\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}$ Kurtosis (see text) Entropy $\gamma\left(1\!-\!\frac{1}{k}\right)+\left(\frac{\lambda}{k}\right)^k +\ln\left(\frac{\lambda}{k}\right)$ mgf see Weibull fading Char. func.

In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function

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

where $x \geq0$ and $k >0$ is the shape parameter and $\lambda >0$ is the scale parameter of the distribution.

The cumulative density function is defined as

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

where again, $x >0$.

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

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

Weibull distributions are often used to model the time until a given technical device fails. If the failure rate of the device decreases over time, one chooses $k<1$ (resulting in a decreasing density $f$). If the failure rate of the device is constant over time, one chooses $k=1$, again resulting in a decreasing function $f$. If the failure rate of the device increases over time, one chooses $k>1$ and obtains a density $f$ which increases towards a maximum and then decreases forever. Manufacturers will often supply the shape and scale parameters for the lifetime distribution of a particular device. The Weibull distribution can also be used to model the distribution of wind speeds at a given location on Earth. Again, every location is characterized by a particular shape and scale parameter.

PropertiesEdit

The nth raw moment is given by:

$m_n = \lambda^n \Gamma(1+n/k)\,$

where $\Gamma$ is the Gamma function. The expected value and standard deviation of a Weibull random variable can be expressed as:

$\textrm{E}(X) = \lambda \Gamma(1+1/k)\,$

and

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

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 kurtosis excess 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:

$\gamma_2=\frac{\lambda^4\Gamma\left(1+\frac{4}{k}\right) -3\sigma^4-4\gamma_1\sigma^3\mu-6\sigma^2\mu^2-\mu^4}{\sigma^4}$

Generating Weibull-distributed random variates Edit

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.

Related distributionsEdit

• $X \sim \mathrm{Exponential}(\lambda)$ is an exponential distribution if $X \sim \mathrm{Weibull}(\gamma = 1, \lambda^{-1})$.
• $X \sim \mathrm{Rayleigh}(\beta)$ is a Rayleigh distribution if $X \sim \mathrm{Weibull}(\gamma = 2, \sqrt{2} \beta)$.
• $\lambda(-\ln(X))^{1/k}\,$ is a Weibull distribution if $X \sim \mathrm{Uniform}(0,1)$.

UsesEdit

The Weibull distribution gives the distribution of lifetimes of objects. It is also used in analysis of systems involving a weakest link. The Weibull distribution is often used in place of the Normal distribution due to the fact that a Weibull variate can be generated through inversion, while Normal variates are typically generated using the more complicated Box-Muller Method, which requires two uniform random variates. Weibull distributions may also be used to represent manufacturing and delivery times in industrial engineering problems, while it is very important in extreme value theory and weather forecasting. It is also a very popular statistical model in reliability engineering and failure analysis, while it is widely applied in radar systems to model the dispersion of the received signals level produced by some types of clutters. Furthermore, concerning wireless communications, the Weibull distribution may be used for fading channel modelling, since the Weibull fading model seems to exhibit good fit to experimental fading channel measurements. The Weibull distribution is also commonly used to describe wind speed distributions as the natural distribution often matches the Weibull shape.