q-exponential distribution
Probability density function
Parameters	shape (real) ${\displaystyle \lambda >0}$rate (real)
Support	${\displaystyle x\in [0,\mathrm{\infty })\text{ for }q\ge 1}$
PDF	${\displaystyle (2-q)\lambda e_q(-\lambda x)}$
CDF	${\displaystyle 1-e_{q^{\prime }}^{-\lambda x/q^{\prime }}\text{ where }{q}^{\prime }=\frac{1}{2-q}}$
Mean	Otherwise undefined
Median	${\displaystyle \frac{-q^{\prime }{\ln }_{q^{\prime }}(1/2)}{\lambda }\text{ where }{q}^{\prime }=\frac{1}{2-q}}$
Mode	0
Variance
Skewness
Excess kurtosis

Theq-exponential distribution is aprobability distribution arising from the maximization of theTsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of aTsallis distribution. Theq-exponential is a generalization of theexponential distribution in the same way that Tsallis entropy is a generalization of standardBoltzmann–Gibbs entropy orShannon entropy.^[1] The exponential distribution is recovered as${\displaystyle q\to 1.}$

Originally proposed by the statisticiansGeorge Box andDavid Cox in 1964,^[2] and known as the reverseBox–Cox transformation for${\displaystyle q=1-\lambda ,}$ a particular case ofpower transform in statistics.

Theq-exponential distribution has the probability density function

${\displaystyle (2-q)\lambda e_q(-\lambda x)}$

where

${\displaystyle e_q(x)=[1+(1-q)x{]}^{1/(1-q)}}$

is theq-exponential ifq ≠ 1. Whenq = 1,e_q(x) is just exp(x).

In a similar procedure to how theexponential distribution can be derived (using the standard Boltzmann–Gibbs entropy or Shannon entropy and constraining the domain of the variable to be positive), theq-exponential distribution can be derived from a maximization of the Tsallis Entropy subject to the appropriate constraints.

Theq-exponential is a special case of thegeneralized Pareto distribution where

${\displaystyle \mu =0,\xi =\frac{q-1}{2-q},\sigma =\frac{1}{\lambda (2-q)}.}$

Theq-exponential is the generalization of theLomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support. The Lomax parameters are:

${\displaystyle \alpha =\frac{2-q}{q-1},{\lambda }_{\mathrm{L}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{1}{\lambda (q-1)}.}$

As the Lomax distribution is a shifted version of thePareto distribution, theq-exponential is a shifted reparameterized generalization of the Pareto. Whenq > 1, theq-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if

${\displaystyle X\sim \mathit{q}\text{-}\mathrm{E}\mathrm{x}\mathrm{p}(q,\lambda )\text{ and }Y\sim \left[\mathrm{Pareto}\left(x_m=\frac{1}{\lambda (q-1)},\alpha =\frac{2-q}{q-1}\right)-x_m\right],}$

then${\displaystyle X\sim Y.}$

Random deviates can be drawn usinginverse transform sampling. Given a variableU that is uniformly distributed on the interval (0,1), then

${\displaystyle X=\frac{-q^{\prime }{\ln }_{q^{\prime }}(U)}{\lambda }\sim \mathit{q}\text{-}\mathrm{E}\mathrm{x}\mathrm{p}(q,\lambda )}$

where${\displaystyle {\ln }_{q^{\prime }}}$ is theq-logarithm and${\displaystyle q^{\prime }=\frac{1}{2-q}.}$

Being apower transform, it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables. It has been found to be an accurate model for train delays.^[3]It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.^[4]

• Constantino Tsallis
• Tsallis statistics
• Tsallis entropy
• Tsallis distribution
• q-copula
• q-Gaussian

• Juniper, J. (2007)"The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty"Archived 2011-07-06 at theWayback Machine, Centre of Full Employment and Equity, The University of Newcastle, Australia

• Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions

• Statistical mechanics
• Continuous distributions
• Probability distributions with non-finite variance

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