# q-Weibull distribution

Parameters Probability density function Cumulative distribution function shape (real) rate (real) shape (real) (see article)

In statistics, the q-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution.

## Characterization

### Probability density function

The probability density function of a q-Weibull random variable is:[1]

where q < 2, > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and

is the q-exponential[1][2][3]

### Cumulative distribution function

The cumulative distribution function of a q-Weibull random variable is:

where

## Mean

The mean of the q-Weibull distribution is

where is the Beta function and is the Gamma function. The expression for the mean is a continuous function of q over the range of definition for which it is finite.

## Relationship to other distributions

The q-Weibull is equivalent to the Weibull distribution when q = 1 and equivalent to the q-exponential when

The q-Weibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support (q < 1) and to include heavy-tailed distributions .

The q-Weibull is a generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support and adds the parameter. The Lomax parameters are:

As the Lomax distribution is a shifted version of the Pareto distribution, the q-Weibull for is a shifted reparameterized generalization of the Pareto. When q > 1, the q-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically: