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 makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent $q$-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.

翻译：Tsallis $q$-高斯分布是标准高斯分布的一种强有力推广，广泛应用于非广延统计力学、金融市场和图像处理等多个领域。它属于由非加性熵刻画的 $q$-分布族。由于 $q$-高斯分布具有灵活性和实用性，在测量模型中成为输入量建模的自然选择。本文给出了独立 $q$-高斯随机变量线性组合的特征函数，并提出了一种数值反演方法。该方法能够在线性测量模型中，当输入量被建模为独立 $q$-高斯随机变量时，准确确定输出量的概率分布。它为基于分布传播进行不确定度分析提供了蒙特卡罗方法之外的一种替代计算途径。