q-exponential distribution

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q-exponential distribution
Probability density function
Probability density plots of q-exponential distributions
Parameters shape (real)
rate (real)
Otherwise undefined
Mode 0
Ex. kurtosis

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.[1] The exponential distribution is recovered as .

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.


Probability density function[edit]

The q-exponential distribution has the probability density function


is the q-exponential, if q≠1. When q=1, eq(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.

Relationship to other distributions[edit]

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:

Generating random deviates[edit]

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 is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.[3]

See also[edit]


  1. ^ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
  2. ^ 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 192611. 
  3. ^ C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution". Phys. Rev. A. 94 (3): 033808. Bibcode:2016PhRvA..94c3808S. doi:10.1103/PhysRevA.94.033808. 

Further reading[edit]

External links[edit]