The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints.
The q-Gaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning.
[citation needed] The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < q < 3.
the q-Gaussian distribution is the PDF of a bounded random variable.
This makes in biology and other domains[2] the q-Gaussian distribution more suitable than Gaussian distribution to model the effect of external stochasticity.
A generalized q-analog of the classical central limit theorem[3] was proposed in 2008, in which the independence constraint for the i.i.d.
variables is relaxed to an extent defined by the q parameter, with independence being recovered as q → 1.
[4] In the heavy tail regions, the distribution is equivalent to the Student's t-distribution with a direct mapping between q and the degrees of freedom.
The choice of the q-Gaussian form may arise if the system is non-extensive, or if there is lack of a connection to small samples sizes.
The standard q-Gaussian has the probability density function [3] where is the q-exponential and the normalization factor
the q-Gaussian distribution is the PDF of a bounded random variable.
As the hypergeometric function is defined for |z| < 1 but x is unbounded, Pfaff transformation could be used.
While it can be justified by an interesting alternative form of entropy, statistically it is a scaled reparametrization of the Student's t-distribution introduced by W. Gosset in 1908 to describe small-sample statistics.
In Gosset's original presentation the degrees of freedom parameter ν was constrained to be a positive integer related to the sample size, but it is readily observed that Gosset's density function is valid for all real values of ν.
[citation needed] The scaled reparametrization introduces the alternative parameters q and β which are related to ν.
Given a Student's t-distribution with ν degrees of freedom, the equivalent q-Gaussian has with inverse Whenever
, the function is simply a scaled version of Student's t-distribution.
It is sometimes argued that the distribution is a generalization of Student's t-distribution to negative and or non-integer degrees of freedom.
However, the theory of Student's t-distribution extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of ν < 0.
[citation needed] As with many distributions centered on zero, the q-Gaussian can be trivially extended to include a location parameter μ.
The density then becomes defined by The Box–Muller transform has been generalized to allow random sampling from q-Gaussians.
[6] The standard Box–Muller technique generates pairs of independent normally distributed variables from equations of the following form.
The generalized Box–Muller technique can generates pairs of q-Gaussian deviates that are not independent.
In practice, only a single deviate will be generated from a pair of uniformly distributed variables.
The following formula will generate deviates from a q-Gaussian with specified parameter q and
These deviates can be transformed to generate deviates from an arbitrary q-Gaussian by It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q-Gaussian.
[7] The q-Gaussian distribution is also obtained as the asymptotic probability density function of the position of the unidimensional motion of a mass subject to two forces: a deterministic force of the type
(determining an infinite potential well) and a stochastic white noise force
Note that in the overdamped/small mass approximation the above-mentioned convergence fails for
[8] Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere have been interpreted as q-Gaussians.