Weibull distribution

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

<math> f(x|k,\lambda) = (k/\lambda) (x/\lambda)^{(k-1)} e^{-(x/\lambda)^k}\,<math>

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

The cumulative density function is defined as

<math>F(x|k,\lambda) = 1- e^{-(x/\lambda)^k}\,<math>

where again, <math>x >0<math>. 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 <math>k<1<math> (resulting in a decreasing density <math>f<math>). If the failure rate of the device is constant over time, one chooses <math>k=1<math>, again resulting in a decreasing function <math>f<math>. If the failure rate of the device increases over time, one chooses <math>k>1<math> and obtains a density <math>f<math> 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.

The expected value and standard deviation of a Weibull random variable can be expressed in terms of the Gamma function:

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

and

<math>\textrm{var}(X) = \lambda^2[\Gamma(1+2/k) – \Gamma^2(1+1/k)]\,<math>

Related distributions

• <math>X \sim \mathrm{Exponential}(\lambda)<math> is an exponential distribution if <math>X \sim \mathrm{Weibull}(\gamma = 1, \lambda)<math>.
• <math>X \sim \mathrm{Rayleigh}(\beta)<math> is a Rayleigh distribution if <math>X \sim \mathrm{Weibull}(\gamma = 2, \beta)<math>.

External links

• The Weibull distribution (with examples, properties, and calculators).
• The Weibull plot.