The cumulative density function is defined as

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((k + 1) / k) and
- var(X) = λ
^{2}[Γ((k + 2) / k) - Γ^{2}((k + 1) / k)]