The dependent responsibility of random quantities is one of the most important factors in probability theory, which is to significantly transfer the dependence of events. This article discusses the concept of independent random variables, their definitions, properties, and some fundamental results related to independence in probability theory. The independence of random variables plays a crucial role in algebraic probability models, mathematical statistics, and stochastic processes. The paper presents theoretical explanations along with illustrative examples that show how the independence of random variables simplifies the analysis of complex probabilistic systems.

Random variable, probability theory, independence, dependent variables, joint probability, algebra, mathematical expectation, distribution.

Феллер В. Введение в теорию вероятностей и ее приложения, 1,2-том. М.: Мир, 1984.

Севастьянов Б.А. Курс теории вероятностей и математичекой статистики. М.: Наука, 1982.

M. Amini, H.R Nili Sani and A. Bozorgnia, Aspects of negative dependence structures,Communications in Statistics-Theory and Methods, 42, (2013), 907-91.

Noura Obeid, Seifedine Kadry Product Of n Independent Maxwell Random Variables // RT&A. 2021. №2 (62).

Hamedani, G.G. and Volkmer, H.W. (2003): A class of symmetricrandom variables; Comm. Statist. Theory-Methods, Vol. 32, pp. 723-728.

Али Абдуламир Тавфик Аль Ани Статистический подход при определении предела текучести твердотельных пористых материалов // Прикладная математика & Физика. 2018. №2.

Independent random variables https://www.statlect.com/fundamentals-of-probability/independent-random-variables

https://www.researchgate.net/publication/282596029_Independence_of_Random_Variables_and_Some_Applications