Some properties of q-Gaussian distributions

Abstract

In this research article, we introduced the notion of q-probabilty distributions in quantum calculus. We characterized the concept of q-density by connecting it to a probability measure and investigated some of their outstanding properties. In this case, the Transfer theorem was extended in order to compute afterwards the q-moments, q-entropy, q-moment generating function, and q-quantiles. We are also interested in finding the centered q-Gaussian distribution N q ( 0 , σ 2 ) with variance σ 2 . We also proved that this q-distribution belongs to a class of classical discrete distributions. The centered q-Gaussian law N q ( 0 , σ 2 ) is also naturally related to the q-Gaussian distribution N q ( μ , σ 2 ) with mean μ and standard deviation σ. We corroborated that the q-moments of these q-distributions are q-analogs of the moments of classical distributions. Numerical studies demonstrated that N q ( 0 , σ 2 ) interpolates between the classical Uniform and Gaussian distributions when q goes to 0 and 1, respectively. Subsequently, simulation studies for various q parameter values and samples sizes of the Gaussian q-distributions were conducted to demonstrate the effectiveness of the proposed model. Eventually, we provided some pertinent closing remarks and offered new perspectives for future works.