q-Gaussian
Probability density function
Parameters	shape (real) ${\displaystyle \beta >0}$ (real)
Support	${\displaystyle x\in (-\mathrm{\infty };+\mathrm{\infty })}$ for ${\displaystyle x\in \left[\pm \frac{1}{\sqrt{\beta (1-q)}}\right]}$ for
PDF	${\displaystyle \frac{\sqrt{\beta }}{C_q}{e}_q(-\beta x^2)}$
CDF	see text
Mean	, otherwise undefined
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance
Skewness
Excess kurtosis

Theq-Gaussian is a probability distribution arising from the maximization of theTsallis entropy under appropriate constraints. It is one example of aTsallis distribution. Theq-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standardBoltzmann–Gibbs entropy orShannon entropy.^[1] Thenormal distribution is recovered asq → 1.

Theq-Gaussian has been applied to problems in the fields ofstatistical mechanics,geology,anatomy,astronomy,economics,finance, andmachine learning.^{[citation needed]} The distribution is often favored for itsheavy tails in comparison to the Gaussian for 1 <q < 3. For theq-Gaussian distribution is the PDF of a boundedrandom variable. This makes in biology and other domains^[2] theq-Gaussian distribution more suitable than Gaussian distribution to model the effect of external stochasticity. A generalizedq-analog of the classicalcentral limit theorem^[3] was proposed in 2008, in which the independence constraint for thei.i.d. variables is relaxed to an extent defined by theq parameter, with independence being recovered asq → 1. However, a proof of such a theorem is still lacking.^[4]

In the heavy tail regions, the distribution is equivalent to theStudent'st-distribution with a direct mapping betweenq and thedegrees of freedom. A practitioner using one of these distributions can therefore parameterize the same distribution in two different ways. The choice of theq-Gaussian form may arise if the system isnon-extensive, or if there is lack of a connection to small samples sizes.

The standardq-Gaussian has the probability density function^[3]

${\displaystyle f(x)=\frac{\sqrt{\beta }}{C_q}{e}_q(-\beta x^2)}$

where

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

is theq-exponential and the normalization factor${\displaystyle C_q}$ is given by

${\displaystyle C_q=\sqrt{\pi }\text{ for }q=1}$

Note that for theq-Gaussian distribution is the PDF of a boundedrandom variable.

For cumulative density function is^[5]

${\displaystyle F(x)=\frac{1}{2}+\frac{\sqrt{q-1}\mathrm{\Gamma }\left(\frac{1}{q-1}\right)x\sqrt{\beta }{}_2{F}_1\left(\frac{1}{2},\frac{1}{q-1};\frac{3}{2};-(q-1)\beta x^2\right)}{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{3-q}{2(q-1)}\right)},}$

where${\displaystyle {}_2{F}_1(a,b;c;z)}$ is thehypergeometric function. As the hypergeometric function is defined for|z| < 1 butx is unbounded, Pfaff transformation could be used.

For,

Just as thenormal distribution is the maximuminformation entropy distribution for fixed values of the first moment${\displaystyle \mathrm{E}(X)}$ and second moment${\displaystyle \mathrm{E}(X^2)}$ (with the fixed zeroth moment${\displaystyle \mathrm{E}(X^0)=1}$ corresponding to the normalization condition), theq-Gaussian distribution is the maximumTsallis entropy distribution for fixed values of these three moments.

While it can be justified by an interesting alternative form of entropy, statistically it is a scaled reparametrization of theStudent'st-distribution introduced by W. Gosset in 1908 to describe small-sample statistics. In Gosset's original presentation the degrees of freedom parameterν 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 ofν.^[6] The scaled reparametrization introduces the alternative parametersq andβ which are related toν.

Given a Student'st-distribution withν degrees of freedom, the equivalentq-Gaussian has

${\displaystyle q=\frac{\nu +3}{\nu +1}\text{ with }\beta =\frac{1}{3-q}}$

with inverse

${\displaystyle \nu =\frac{3-q}{q-1},\text{ but only if }\beta =\frac{1}{3-q}.}$

Whenever${\displaystyle \beta \ne \frac{1}{3-q}}$, the function is simply a scaled version of Student'st-distribution.

It is sometimes argued that the distribution is a generalization of Student'st-distribution to negative and or non-integer degrees of freedom. However, the theory of Student'st-distribution extends trivially to all real degrees of freedom, where the support of the distribution is nowcompact rather than infinite in the case ofν < 0.

As with many distributions centered on zero, theq-Gaussian can be trivially extended to include a location parameterμ. The density then becomes defined by

${\displaystyle \frac{\sqrt{\beta }}{C_q}{e}_q(-\beta (x-\mu {)}^2).}$

TheBox–Muller transform has been generalized to allow random sampling fromq-Gaussians.^[7][8] The standard Box–Muller technique generates pairs of independent normally distributed variables from equations of the following form.

${\displaystyle Z_1=\sqrt{-2\ln (U_1)}\cos (2\pi U_2)}$

${\displaystyle Z_2=\sqrt{-2\ln (U_1)}\sin (2\pi U_2)}$

The generalized Box–Muller technique can generates pairs ofq-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 aq-Gaussian with specified parameterq and${\displaystyle \beta =\frac{1}{3-q}}$

${\displaystyle Z=\sqrt{-2{\text{ ln}}_{q^{\prime }}(U_1)}\text{ cos}(2\pi U_2)}$

where${\displaystyle {\text{ ln}}_q}$ is theq-logarithm and${\displaystyle q^{\prime }=\frac{1+q}{3-q}}$

These deviates can be transformed to generate deviates from an arbitraryq-Gaussian by

${\displaystyle Z^{\prime }=\mu +\frac{Z}{\sqrt{\beta (3-q)}}}$

It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is aq-Gaussian.^[9]

Theq-Gaussian distribution is also obtained as the asymptoticprobability density function of the position of the unidimensional motion of a mass subject to two forces: a deterministic force of the type$F_1(x)=-2x/(1-x^2)$ (determining an infinite potential well) and a stochastic white noise force$F_2(t)=\sqrt{2(1-q)}\xi (t)$, where${\displaystyle \xi (t)}$ is awhite noise. Note that in the overdamped/small mass approximation the above-mentioned convergence fails for, as recently shown.^[10]

Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere have been interpreted asq-Gaussians.^[11][12]

• Constantino Tsallis
• Tsallis statistics
• Tsallis entropy
• Tsallis distribution
• q-exponential distribution
• Q-Gaussian process
• κ-Gaussian distribution
• generalized Gaussian distribution

• Juniper, J. (2007)"The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty"(PDF). Archived fromthe original(PDF) on 2011-07-06. Retrieved2011-06-24., Centre of Full Employment and Equity, The University of Newcastle, Australia

• Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions

• Statistical mechanics
• Continuous distributions
• Probability distributions with non-finite variance

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