Characteristic Function of the Tsallis $q$-Gaussian and Its Applications in Measurement and Metrology

The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets, and image processing. It belongs to the $q$-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, $q$-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent $q$-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique enables the assessment of the probability distribution of output quantities in linear measurement models and the conduct of uncertainty analysis in metrology.