# 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 $X \sim \mathrm{Uniform}(0,1)$.