qWeibull distribution
Probability density function  
Cumulative distribution function  
Parameters 
shape (real) rate (real) shape (real) 

Support 

CDF  
Mean  (see article) 
In statistics, the qWeibull 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 qWeibull random variable is:^{[1]}
where q < 2, > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and
is the qexponential^{[1]}^{[2]}^{[3]}
Cumulative distribution function
The cumulative distribution function of a qWeibull random variable is:
where
Mean
The mean of the qWeibull 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 qWeibull is equivalent to the Weibull distribution when q = 1 and equivalent to the qexponential when
The qWeibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support (q < 1) and to include heavytailed distributions .
The qWeibull 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 qWeibull for is a shifted reparameterized generalization of the Pareto. When q > 1, the qexponential is equivalent to the Pareto shifted to have support starting at zero. Specifically:
See also
References
 1 2 Picoli, S. Jr.; Mendes, R. S.; Malacarne, L. C. (2003). "qexponential, Weibull, and qWeibull distributions: an empirical analysis". arXiv:condmat/0301552.
 ↑ Naudts, Jan (2010). "The qexponential family in statistical physics" (PDF). J. Phys. Conf. Ser. IOP Publishing. 201. doi:10.1088/17426596/201/1/012003. Retrieved 9 June 2014.
 ↑ "On a qCentral Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan J. Math. 76. 2008. doi:10.1007/s000320080087y. Retrieved 9 June 2014.