# Weibull distribution

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

[itex] f(x|k,\lambda) = (k/\lambda) (x/\lambda)^{(k-1)} e^{-(x/\lambda)^k}\,[itex]

where [itex]x >0[itex] and [itex]k >0[itex] is the shape parameter and [itex]\lambda >0[itex] is the scale parameter of the distribution.

The cumulative density function is defined as

[itex]F(x|k,\lambda) = 1- e^{-(x/\lambda)^k}\,[itex]

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

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

and

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

## Related distributions

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

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