Probability density function

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

${\displaystyle f(x)={{\sqrt {\beta }} \over C_{q}}e_{q}(-\beta x^{2})}$

where

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

is the q-exponential and the normalization factor ${\displaystyle C_{q}}$ is given by

${\displaystyle C_{q}={{2{\sqrt {\pi }}\Gamma \left({1 \over 1-q}\right)} \over {(3-q){\sqrt {1-q}}\Gamma \left({3-q \over 2(1-q)}\right)}}{\text{ for }}-\infty <q<1}$

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

${\displaystyle C_{q}={{{\sqrt {\pi }}\Gamma \left({3-q \over 2(q-1)}\right)} \over {{\sqrt {q-1}}\Gamma \left({1 \over q-1}\right)}}{\text{ for }}1<q<3.}$

Note that for ${\displaystyle q<1}$ the q-Gaussian distribution is the PDF of a bounded random variable.

Cumulative density function

For ${\displaystyle 1<q<3}$ cumulative density function is ^[5]

${\displaystyle F(x)={\frac {1}{2}}+{\frac {{\sqrt {q-1}}\,\Gamma \left({1 \over q-1}\right)x{\sqrt {\beta }}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{q-1}};{\tfrac {3}{2}};-(q-1)\beta x^{2}\right)}{{\sqrt {\pi }}\,\Gamma \left({3-q \over 2(q-1)}\right)}},}$

where ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$ is the hypergeometric function. As the hypergeometric function is defined for |z| < 1 but x is unbounded, Pfaff transformation could be used.

For ${\displaystyle q<1}$,

$${\displaystyle F(x)={\begin{cases}0&x<-{\frac {1}{\sqrt {\beta (1-q)}}},\\{\frac {1}{2}}+{\frac {{\sqrt {1-q}}\,\Gamma \left({5-3q \over 2(q-1)}\right)x{\sqrt {\beta }}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{q-1}};{\tfrac {3}{2}};-(q-1)\beta x^{2}\right)}{{\sqrt {\pi }}\,\Gamma \left({2-q \over 1-q}\right)}}&-{\frac {1}{\sqrt {\beta (1-q)}}}<x<{\frac {1}{\sqrt {\beta (1-q)}}},\\1&x>{\frac {1}{\sqrt {\beta (1-q)}}}.\end{cases}}}$$

Student's t-distribution

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. 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

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

with inverse

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

Whenever ${\displaystyle \beta \neq {1 \over {3-q}}}$, 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.

Three-parameter version

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

${\displaystyle {{\sqrt {\beta }} \over C_{q}}e_{q}({-\beta (x-\mu )^{2}}).}$

Physics

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 ${\textstyle F_{1}(x)=-2x/(1-x^{2})}$ (determining an infinite potential well) and a stochastic white noise force ${\textstyle F_{2}(t)={\sqrt {2(1-q)}}\xi (t)}$, where ${\displaystyle \xi (t)}$ is a white noise. Note that in the overdamped/small mass approximation the above-mentioned convergence fails for ${\displaystyle q<0}$, as recently shown.^[8]

Finance

Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere have been interpreted as q-Gaussians.^[9][10]