en>Qetuth: more specific stub type

2012-01-01T02:18:39Z

more specific stub type

New page

{{DISPLAYTITLE:q-exponential distribution}}
{{Probability distribution |
name =q-exponential distribution|
type =density|
pdf_image =[[File:Qexponential.png|325px|Probability density plots of q-exponential distributions]]|
parameters =<math>q < 2 </math> [[shape parameter|shape]] ([[Real number|real]]) <br /> <math> \lambda > 0 </math> [[rate parameter|rate]] ([[Real number|real]]) |
support =<math>x \in [0; +\infty)\! \text{ for }q \ge 1 </math> <br /> <math> x \in [0; {1 \over {\lambda(1-q)}}) \text{ for } q<1 </math>|
pdf =<math>{ (2-q) \lambda e_q^{-\lambda x}} </math>|
cdf =<math>{ 1-e_{q'}^{-\lambda x \over q'}} \text{ where } q' = {1 \over {2-q}}</math>|
mean =<math>{ 1 \over \lambda (3-2q) } \text{ for }q < {3 \over 2} </math> <br /> Otherwise undefined|
median =<math>{ {-q' \text{ ln}_{q'}({1 \over 2})} \over {\lambda}} \text{ where } q' = {1 \over {2-q}}</math>|
mode =0|
variance =<math>{{ q-2 } \over { (2q-3)^2 (3q-4) \lambda^2}} \text{ for }q < {4 \over 3}</math>|
skewness =<math> {2 \over {5-4q}} \sqrt{{3q-4} \over {q-2} } \text{ for }q < {5 \over 4}</math>|
kurtosis =<math>6{{ -4q^3 + 17q^2 - 20q + 6 } \over { (q-2) (4q-5) (5q-6) }} \text{ for }q < {6 \over 5}</math>|
entropy =|
mgf =|
cf =|
}}

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 [[Entropy (statistical thermodynamics)|Boltzmann–Gibbs entropy]] or [[Entropy (information theory)|Shannon Entropy]].<ref>Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356</ref> The [[exponential distribution]] is recovered as <math>q \rightarrow 1</math>.

==Characterization==

===Probability density function===
The q-exponential distribution has the probability density function

:<math>{ (2-q) \lambda e_q^{-\lambda x}} </math>

where

:<math>e_q(x) = [1+(1-q)x]^{1 \over 1-q}</math>

is the [[Tsallis statistics#q-exponential|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
:<math> \mu = 0 ~,~ \xi = {{q-1} \over {2-q}} ~,~ \sigma = {1 \over {\lambda (2-q)}}</math>

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:
:<math> \alpha = { {2-q} \over {q-1}} ~,~ \lambda_{lomax} = {1 \over {\lambda (q-1)}} </math>

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:
:<math>
\text{If } X \sim \mbox{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 \,
</math>

==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
:<math>
X = {{-q' \text{ ln}_{q'}(U)} \over \lambda} \sim \mbox{qExp}(q,\lambda)</math>
where <math>\text{ln}_{q'}</math> is the [[Tsallis statistics#q-logarithm|q-logarithm]] and <math> q' = {1 \over {2-q}}</math>

== Applications ==

=== Economics (econophysics) ===

The q-exponential distribution has been used to describe the distribution of wealth (assets) between individuals.<ref>Adrian A. Dragulescu [http://arxiv.org/abs/cond-mat/0307341 Applications of physics to economics and finance: Money, income, wealth, and the stock market] arXiv:cond-mat/0307341v2</ref>

== See also ==

* [[Constantino Tsallis]]
* [[Tsallis statistics]]
* [[Tsallis entropy]]
* [[Tsallis distribution]]
* [[q-Gaussian]]

== Notes ==

{{reflist}}

==Further reading==
*Juniper, J. (2007) [http://e1.newcastle.edu.au/coffee/pubs/wp/2007/07-10.pdf "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

== External links ==
* [http://www.cscs.umich.edu/~crshalizi/notebooks/tsallis.html Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions]

{{Tsallis}}
{{ProbDistributions|continuous-variable}}

{{DEFAULTSORT:Q-Exponential Distribution}}
[[Category:Statistical mechanics]]
[[Category:Continuous distributions]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Probability distributions]]

Gradient-like vector field - Revision history

en>Qetuth: more specific stub type