|Probability density function
In q-analog theory, 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 .
Probability density function
The q-exponential distribution has the probability density function
is the q-exponential.
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
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
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
The q-exponential distribution has been used to describe the distribution of wealth (assets) between individuals.
See also↑Jump back a section
- Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
- Adrian A. Dragulescu Applications of physics to economics and finance: Money, income, wealth, and the stock market arXiv:cond-mat/0307341v2
- 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