The study of probability distributions for random variables and their algebraic combinations has been a central focus driving the advancement of probability and statistics. Since the 1920s, the challenge of calculating the probability distributions of sums, differences, products, and quotients of independent random variables have drawn the attention of numerous statisticians and mathematicians who studied the algebraic properties and relationships of random variables. Statistical distributions are highly helpful in data science and machine learning, as they provide a range of possible values for the variables, aiding in the development of a deeper understanding of the underlying problem. In this paper, we have presented a new probability distribution based on the $\hat{I}$-function. Also, we have discussed the applications of the $\hat{I}$ function, particularly in deriving the distributions of product and the quotient involving two independent $\hat{I}$ function variates. Additionally, it has been shown that both the product and quotient of two independent $\hat{I}$-function variates also follow the $\hat{I}$-function distribution. Furthermore, the new distribution, known as the $\hat{I}$-function distribution, includes several well-known classical distributions such as the gamma, beta, exponential, normal H-function, and G-function distributions, among others, as special cases. Therefore, the $\hat{I}$-function distribution can be considered a characterization or generalization of the above-mentioned distributions.

翻译：随机变量及其代数组合的概率分布研究，一直是推动概率论与统计学发展的核心焦点。自二十世纪二十年代以来，计算独立随机变量之和、差、积、商的概率分布这一挑战，吸引了众多统计学家和数学家的关注，他们研究了随机变量的代数性质与关系。统计分布在数据科学与机器学习中极具价值，因其提供了变量可能取值的范围，有助于深化对潜在问题的理解。本文提出了一种基于$\hat{I}$函数的新概率分布。同时，我们探讨了$\hat{I}$函数的应用，特别是在推导涉及两个独立$\hat{I}$函数变量的乘积与商分布方面。此外，研究证明两个独立$\hat{I}$函数变量的乘积与商同样服从$\hat{I}$函数分布。进一步地，这一被称为$\hat{I}$函数分布的新分布，将多种经典分布如伽马分布、贝塔分布、指数分布、正态H函数分布及G函数分布等纳入其特例范畴。因此，$\hat{I}$函数分布可视为上述分布的表征或推广。