Gaussian q-distribution

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This article is about the distribution introduced by Diaz and Teruel. For the Tsallis q-Gaussian, see q-Gaussian.

In mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel,[clarification needed] is a q-analogue of the Gaussian or normal distribution.

The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.

Definition[edit]

The Gaussian q-density.

Let q be a real number in the interval [0, 1). The probability density function of the Gaussian q-distribution is given by

s_q(x) = \begin{cases} 0 & \text{if }  x < -\nu  \\ \frac{1}{c(q)}E_{q^2}^{\frac{-q^2x^2}{[2]_q}}  & \text{if } -\nu \leq x \leq \nu \\ 0 & \mbox{if } x >\nu. \end{cases}

where

\nu = \nu(q) = \frac{1}{\sqrt{1-q}} ,
c(q)=2(1-q)^{1/2}\sum_{m=0}^\infty \frac{(-1)^m q^{m(m+1)}}{(1-q^{2m+1})(1-q^2)_{q^2}^m} .

The q-analogue [t]q of the real number  t is given by

 [t]_q=\frac{q^t-1}{q-1}.

The q-analogue of the exponential function is the q-exponential, Ex
q
, which is given by

 E_q^{x}=\sum_{j=0}^{\infty}q^{j(j-1)/2}\frac{x^{j}}{[j]!}

where the q-analogue of the factorial is the q-factorial, [n]q!, which is in turn given by

 [n]_q!=[n]_q[n-1]_q\cdots [2]_q \,

for an integer n > 2 and [1]q! = [0]q! = 1.

The Cumulative Gaussian q-distribution.

The cumulative distribution function of the Gaussian q-distribution is given by

G_q(x) = \begin{cases} 0 & \text{if } x < -\nu \\[12pt]
\displaystyle \frac{1}{c(q)}\int_{-\nu}^{x} E_{q^2}^{-q^2 t^2/[2]} \, d_qt & \text{if }  -\nu \leq x \leq \nu \\[12pt]
1 & \text{if } x>\nu
\end{cases}

where the integration symbol denotes the Jackson integral.

The function Gq is given explicitly by

G_q(x)= \begin{cases}  0 & \text{if } x < -\nu, \\
\displaystyle \frac{1}{2} + \frac{1-q}{c(q)} \sum_{n=0}^\infty \frac{q^{n(n+1)}(q-1)^n}{(1-q^{2n+1})(1-q^2)_{q^2}^{n}}x^{2n+1} &  \text{if } -\nu \leq x \leq \nu \\
1 & \text{if}\ x > \nu
\end{cases}

where

(a+b)_q^n=\prod_{i=0}^{n-1}(a+q^ib) .

Moments[edit]

The moments of the Gaussian q-distribution are given by

\frac{1}{c(q)}\int_{-\nu}^\nu E_{q^2}^{-q^2 x^2/[2]} \, x^{2n} \, d_qx =[2n-1]!! ,
\frac{1}{c(q)}\int_{-\nu}^\nu E_{q^{2}}^{-q^2 x^2/[2]} \, x^{2n+1} \, d_qx=0 ,

where the symbol [2n − 1]!! is the q-analogue of the double factorial given by

 [2n-1][2n-3]\cdots[1]= [2n-1]!!. \,

References[edit]