# q-exponential distribution

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Parameters Probability density function $q < 2$ shape (real) $\lambda > 0$ rate (real) $x \in [0; +\infty)\! \text{ for }q \ge 1$ $x \in [0; {1 \over {\lambda(1-q)}}) \text{ for } q<1$ ${ (2-q) \lambda e_q^{-\lambda x}}$ ${ 1-e_{q'}^{-\lambda x \over q'}} \text{ where } q' = {1 \over {2-q}}$ ${ 1 \over \lambda (3-2q) } \text{ for }q < {3 \over 2}$ Otherwise undefined ${ {-q' \text{ ln}_{q'}({1 \over 2})} \over {\lambda}} \text{ where } q' = {1 \over {2-q}}$ 0 ${{ q-2 } \over { (2q-3)^2 (3q-4) \lambda^2}} \text{ for }q < {4 \over 3}$ ${2 \over {5-4q}} \sqrt{{3q-4} \over {q-2} } \text{ for }q < {5 \over 4}$ $6{{ -4q^3 + 17q^2 - 20q + 6 } \over { (q-2) (4q-5) (5q-6) }} \text{ for }q < {6 \over 5}$

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.[1] The exponential distribution is recovered as $q \rightarrow 1$.

## Characterization

### Probability density function

The q-exponential distribution has the probability density function

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

where

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

is the q-exponential.

## Derivation

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

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

$\mu = 0 ~,~ \xi = {{q-1} \over {2-q}} ~,~ \sigma = {1 \over {\lambda (2-q)}}$

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:

$\alpha = { {2-q} \over {q-1}} ~,~ \lambda_\text{Lomax} = {1 \over {\lambda (q-1)}}$

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:

$\text{If } X \sim \mathrm{qExp}(q,\lambda) \text{ and } Y \sim \left[\text{Pareto} \left( x_m = {1 \over {\lambda (q-1)}}, \alpha = { {2-q} \over {q-1}} \right) -x_m \right], \text{ then } X \sim Y \,$

## Generating random deviates

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

$X = {{-q' \text{ ln}_{q'}(U)} \over \lambda} \sim \mbox{qExp}(q,\lambda)$

where $\text{ln}_{q'}$ is the q-logarithm and $q' = {1 \over {2-q}}$

## Applications

### Economics (econophysics)

The q-exponential distribution has been used to describe the distribution of wealth (assets) between individuals.[2]

## Notes

1. ^ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
2. ^ Adrian A. Dragulescu Applications of physics to economics and finance: Money, income, wealth, and the stock market arXiv:cond-mat/0307341v2