q-exponential distribution

The q-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The q-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as ${\displaystyle q\rightarrow 1.}$

q-exponential distribution

Probability density function
Parameters	${\displaystyle q<2}$ shape (real) ${\displaystyle \lambda >0}$ rate (real)
Support	${\displaystyle x\in [0,\infty ){\text{ for }}q\geq 1}$ ${\displaystyle x\in \left[0,{\frac {1}{\lambda (1-q)}}\right){\text{ for }}q<1}$
PDF	${\displaystyle (2-q)\lambda e_{q}^{-\lambda x}}$
CDF	${\displaystyle 1-e_{q'}^{-\lambda x/q'}{\text{ where }}q'={\frac {1}{2-q}}}$
Mean	${\displaystyle {\frac {1}{\lambda (3-2q)}}{\text{ for }}q<{\frac {3}{2}}}$ Otherwise undefined
Median	${\displaystyle {\frac {-q'\ln _{q'}(1/2)}{\lambda }}{\text{ where }}q'={\frac {1}{2-q}}}$
Mode	0
Variance	${\displaystyle {\frac {q-2}{(2q-3)^{2}(3q-4)\lambda ^{2}}}{\text{ for }}q<{\frac {4}{3}}}$
Skewness	${\displaystyle {\frac {2}{5-4q}}{\sqrt {\frac {3q-4}{q-2}}}{\text{ for }}q<{\frac {5}{4}}}$
Ex. kurtosis	${\displaystyle 6{\frac {-4q^{3}+17q^{2}-20q+6}{(q-2)(4q-5)(5q-6)}}{\text{ for }}q<{\frac {6}{5}}}$

Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for ${\displaystyle q=1-\lambda ,}$ a particular case of power transform in statistics.