Q-Gaussian distribution

q-Gaussian

Probability density function
Parameters	${\displaystyle q<3}$ shape (real) ${\displaystyle \beta >0}$ (real)
Support	${\displaystyle x\in (-\infty ;+\infty )\!}$ for ${\displaystyle 1\leq q<3}$ ${\displaystyle x\in \left[\pm {1 \over {\sqrt {\beta (1-q)}}}\right]}$ for ${\displaystyle q<1}$
PDF	${\displaystyle {{\sqrt {\beta }} \over C_{q}}e_{q}({-\beta x^{2}})}$
CDF	see text
Mean	${\displaystyle 0{\text{ for }}q<2}$, otherwise undefined
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle {1 \over {\beta (5-3q)}}{\text{ for }}q<{5 \over 3}}$ ${\displaystyle \infty {\text{ for }}{5 \over 3}\leq q<2}$ ${\displaystyle {\text{Undefined for }}2\leq q<3}$
Skewness	${\displaystyle 0{\text{ for }}q<{3 \over 2}}$
Excess kurtosis	${\displaystyle 6{q-1 \over 7-5q}{\text{ for }}q<{7 \over 5}}$

The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy.^[1] The normal distribution is recovered as q → 1.

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. For ${\displaystyle q<1}$ 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. However, a proof of such a theorem is still lacking.^[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. A practitioner using one of these distributions can therefore parameterize the same distribution in two different ways. 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.