Integral Representation of Cumulative Distribution of Random Variables and Its Statistical Applications

Abstract

The integral representation of the cumulative distribution function is a core tool in probability theory and statistics. By integrating probability densities, it enables precise quantification of random phenomena. It spans the foundations of calculus, from Newton and Leibniz to Cauchy's rigorous limit formulation, and to its extension to fractional integrals, demonstrating the evolution and depth of probability theory. This framework, through interval probabilities, expected value calculations, and fractional derivatives, unifies the description of continuous and discrete random variables, supporting improved precision in parameter estimation and hypothesis testing in statistical inference. It has driven quantitative breakthroughs in option pricing and risk management in the financial sector, and, through memory modeling, captures the turbulence of network dynamics in complex systems, long-range correlations in climate change and biological systems, revealing the dynamic behavior of history dependence. Its universality and rigor make it a bridge for interdisciplinary applications, deepening the theoretical and practical value of probability theory in statistics, finance, and complex systems analysis.