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Does anyone know what is the Cumulative distribution function of a q-Gaussian distribution?

The pdf can be found online, as the Wikipedia article on this topic, but unlike most articles on probability distributions, a CDF is not given.

Does anyone know if the functional form of the CDF exists, and if so what it is?

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  • $\begingroup$ Yes, the CDF exists. I will denote it $\Phi_{q,\beta}(x)$. For a given $q < 3$ and $\beta > 0$ it provides the cumulative distribution of the q-Gaussian with parameters q and $\beta$, evaluated at x. It exists every bit as much as sin(x), $\Gamma(x)$ or the standard Normal cdf,, $\Phi(x)$. As for this function's absence on calculators, and various libraries and software, that is just because no one happened to implement the function there yet, based on obscurity and lack of demand. Nowadays, many people do view $\Phi(x)$, as being a closed form. $\endgroup$
    – Mark L. Stone
    Commented Jun 5, 2018 at 17:10
  • $\begingroup$ @MarkL.Stone what about "the functional form of the CDF" as in the question? $\endgroup$
    – AlexTP
    Commented Jun 5, 2018 at 21:35
  • $\begingroup$ @AlexTP I provided the "functional form", it's $\Phi_{q,\beta}(x)$, just as sin(x) is the functional form of sin(x). And standing behind that is the ability to numerically compute its value to arbitrarily high accuracy, given high enough precision (digits) in the calculation. Beyond that, it's just a matter of fashion and common availability, or not, of this function. $\endgroup$
    – Mark L. Stone
    Commented Jun 5, 2018 at 21:40
  • $\begingroup$ The Wikipedia article correctly notes that the q-Gaussian is a reparameterization of Student t distributions (for $q\gt 1;$ for $q\le 1,$ it extends the Student t family). Thus, like the Student t, the CDF can be expressed in terms of the incomplete Beta function (which in turn has an expression in terms of the hypergeometric function $_2F_1$). Contrary to the assertion in this question, that article does give the CDF (in terms of the Riemann hypergeometric function). $\endgroup$
    – whuber ♦
    Commented Apr 16, 2022 at 20:22

2 Answers 2

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Since we don't have a closed form for the CDF of the Gaussian distribution (other than the integral over the PDF), and the q-Gaussian is a generalization, we know that at least for $q\to1$, there is a q-Gaussian that doesn't have a closed-form CDF.

So, no general CDF for you, sorry.

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  • $\begingroup$ If that's so, then there's no CDF for Student's t distribution, either! $\endgroup$
    – whuber ♦
    Commented Aug 29, 2023 at 22:25
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It depends on what you mean by a "closed form" as a closed form can be in the eye of the beholder.

Mathematica has a TsallisQGaussianDistribution function which has 3 parameters. To match the Wiki definition which has parameters $\beta$ and $q$ those 3 parameters are $0$, $1/\sqrt{2\beta}$, and $q$.

The CDF is given by the following:

CDF of q Gaussian distribution

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  • $\begingroup$ This simplifies considerably when expressed in terms of regularized incomplete Beta functions. Relating these formulas to the CDF of a Student t distribution would be more insightful (and useful to those using standard statistical programming platforms). $\endgroup$
    – whuber ♦
    Commented Apr 16, 2022 at 20:23
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    $\begingroup$ @whuber Thanks! I will work on that and update the answer. $\endgroup$
    – JimB
    Commented Apr 17, 2022 at 0:59

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