New q-general natural exponential family

This paper explores the extended q-natural exponential family (q-NEF) from a probabilistic standpoint. We introduce this family within the framework of q-calculus, generalizing the classical natural exponential family by replacing the traditional exponential kernel eθx with the q-kernel eqθxf(x)q−1. The study examines both continuous and discrete distributions and investigates the properties of the q-NEF, inspired by concepts from statistical mechanics. The paper also establishes the q-Laplace transform for probability distributions. We explicitly calculate the q-NEFs for several distributions, including the q-Bernoulli, q-Binomial, q-Exponential. Furthermore, a new generalization of the q-Exponential distribution (q-GE) is introduced, which is important for stress-strength reliability modeling and lifetime data analysis. To evaluate the efficacy of these q-distributions, we carry out simulation studies, offering valuable insights into their performance.