Q-exponential distribution

q-exponential distribution

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

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 ${\displaystyle q\rightarrow 1}$.

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.