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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

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Originally proposed by the statisticians George Box and David Cox in 1964,^{[2]} and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.

The *q*-exponential distribution has the probability density function

where

is the *q*-exponential if *q* ≠ 1. When *q* = 1, *e*_{q}(x) is just exp(*x*).

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

The *q*-exponential is a special case of the generalized Pareto distribution where

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

As the Lomax distribution is a shifted version of the Pareto distribution, the *q*-exponential 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, if

then

Random deviates can be drawn using inverse transform sampling. Given a variable *U* that is uniformly distributed on the interval (0,1), then

where is the *q*-logarithm and

Being a power 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]}

**^**Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356**^**Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations".*Journal of the Royal Statistical Society, Series B*.**26**(2): 211–252. JSTOR 2984418. MR 0192611.**^**Keith Briggs and Christian Beck (2007). "Modelling train delays with*q*-exponential functions".*Physica A*.**378**(2): 498–504. arXiv:physics/0611097. Bibcode:2007PhyA..378..498B. doi:10.1016/j.physa.2006.11.084. S2CID 107475.**^**C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution".*Phys. Rev. A*.**94**(3): 033808. arXiv:1606.08430. Bibcode:2016PhRvA..94c3808S. doi:10.1103/PhysRevA.94.033808. S2CID 119317114.

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

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