A Comprehensive study of the Central Limit Theorem and its impact on Statistical Modelling

Abstract

A fundamental concept in probability theory and statistical inference, the Central Limit Theorem (CLT) offers an effective structure for comprehending the behavior of sums of independent random variables. This paper provides an in-depth analysis of the CLT, exploring its mathematical proofs, theoretical underpinnings, and central hypotheses. Many different statistical techniques and methodologies are based on the theorem’s ability to estimate the distribution of sample means to a normal distribution, regardless of the actual distribution of the data. Confidence interval estimates, hypothesis testing, and trustworthy statistical inference are made possible by the crucial role that the CLT plays in permitting both large and small sample scenarios. The study emphasizes the practical applications of the CLT in domains including finance, engineering, data science, and the natural sciences, in addition to its theoretical significance. The theorem’s adaptability and wide-ranging influence are demonstrated by specific case studies that highlight its application in large-scale data analysis, quality control procedures, and stock price modelling. The flexibility of the CLT in increasingly complicated situations, such as dependent data and non-identically distributed variables, is proven by looking at extensions and modifications of the CLT, such as the Lindeberg and Lyapunov conditions. This work emphasizes the mathematical rigor and practical value of the CLT across several domains, offering a comprehensive knowledge of its far-reaching repercussions for statistical modeling.