Probability density function Cumulative distribution function Parameters scale (real) shape (real) Support pdf cdf Mean Median Mode Variance Skewness Kurtosis (see text) Entropy 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 where and is the shape parameter and is the scale parameter of the distribution.

The cumulative density function is defined as where again, .

The failure rate h (or hazard rate) is given by: 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 (resulting in a decreasing density ). If the failure rate of the device is constant over time, one chooses , again resulting in a decreasing function . If the failure rate of the device increases over time, one chooses and obtains a density 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.

## Properties

The nth raw moment is given by: where is the Gamma function. The expected value and standard deviation of a Weibull random variable can be expressed as: and The skewness is given by: The kurtosis excess is given by: where . The kurtosis excess may also be written: ## Generating Weibull-distributed random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1], then the variate has a Weibull distribution with parameters k and λ. This follows from the form of the cumulative distribution function.

## Related distributions

• is an exponential distribution if .
• is a Rayleigh distribution if .
• is a Weibull distribution if .