Weibull distribution

<math> f(x) = \frac{k x^{k-1}}{\lambda^k} e^{-(x/\lambda)^k} \qquad \mbox{for } x>0</math>

where k >0 is the shape parameter and λ > 0 is the scale parameter of the distribution.

The Exponential distribution (when k = 1) and Rayleigh distribution (when k = 2) are two special cases of the Weibull distribution.

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.

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

E(X) = λ Γ(1 + 1 / k) and

var(X) = λ²[Γ(1 + 2 / k) - Γ²(1 + 1 / k)]