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Abstract

Abstract There arenindependent identically distributed (i.i.d.) uniform random variables defined as ξ1,ξ2,…,ξn, valued in interval [0,k](k>0), and there is a constantmvalued in interval (0,kn). We study the distribution of the product of these random numbers and prove a formula calculating Pr(∏i=1nξi≤m). Interestingly, we find that the result is exactly the sum of the firstnterms in the Taylor series expansion of the function exp(x) with x=nlnk-lnm. Through considering the corresponding probability density function, we make an extension of the formula calculating Pr(∏i=1nξi≤m) to any positive realn, and the extended formula can be written in a concise form of regularized gamma function. Then we discuss the probability in the limit n→∞ with m=1 and prove that it tends to 1,1/2 and 0 when k∈(0,e), k=e and k∈(e,∞), respectively.

Author Bio

Suwen Tian, a graduate at Beijing No. 8 High School and about to enter the school of mathematics of Tianjin University as an undergraduate, is 16 years old this year and has currently completed the third year of high school. He has had a passion for exploring mathematics since childhood, extensively diving into various popular science and professional books. Presently, he has independently studied and mastered the fundamental content of college-level calculus, linear algebra, and group theory through self-learning. Additionally, he possesses a comprehensive understanding of deeper mathematical concepts such as the gamma function. Drawing from his extracurricular time, he has authored two mathematical papers, including the present one, encompassing topics in number theory, probability theory, and analysis. His aspiration is to tackle numerous challenging problems within mathematics, and he has already commenced researching and working on certain mathematical topics proposed by himself and previous scholars. He aims to persistently cultivate his expertise in the field of mathematics in the long term and is dedicated to achieve this goal.

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