两个独立$\hat{I}$函数变量的乘积与商之分布
On the Distributions of Product and Quotient of two Independent $\hat{I}$-function variates
AI总结：
本文提出基于$\hat{I}$函数的新概率分布，证明两个独立$\hat{I}$函数变量的乘积与商仍服从该分布，并涵盖伽马、贝塔、指数、H函数和G函数等经典分布作为特例，实现统一推广。
摘要：
对随机变量及其代数组合的概率分布的研究一直是推动概率论与统计学发展的核心焦点。自20世纪20年代以来，计算独立随机变量的和、差、积与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相互关系。统计分布在数据科学和机器学习中极为有用，因为它们提供了变量的可能取值范围，有助于更深入地理解潜在问题。本文基于$\hat{I}$函数提出了一种新的概率分布。同时，我们讨论了$\hat{I}$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的分布方面。此外，我们证明了两个独立$\hat{I}$函数变量的乘积与商也服从$\hat{I}$函数分布。进一步地，这一新分布，即$\hat{I}$函数分布，将若干经典分布如伽马分布、贝塔分布、指数分布、正态H函数分布和G函数分布等作为特例包含在内。因此，$\hat{I}$函数分布可被视为上述分布的一种特征化或推广。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
