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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 Jun 5 '18 at 17:10
  • $\begingroup$ @MarkL.Stone what about "the functional form of the CDF" as in the question? $\endgroup$ – AlexTP Jun 5 '18 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 Jun 5 '18 at 21:40
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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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