Poisson Distribution and its Relationship to The Normal and Binomial Distributions: Review Article
Abstract
The Poisson Distribution (PD) is a foundational statistical model widely utilized in probability theory to represent the frequency of discrete, independent events within a fixed interval of time or space. Its analytical structure, based on the parameter λ, allows it to effectively model rare occurrences in diverse fields such as telecommunications, health, commerce, and environmental science. It also serves as a mathematical bridge between the Binomial distribution under Bernoulli trials and the Normal distribution under large-sample conditions. Despite the PD’s established applications, a comprehensive synthesis of its convergence behavior and comparative properties with the Binomial and Normal distributions remains underexplored in the literature. This article aims to review the mathematical relationships and convergence properties of the PD, particularly in the context of approximations to Binomial and Normal distributions, and to reaffirm its applicability in modeling real-world phenomena. The analysis confirms that the PD approximates the Binomial distribution when the number of trials is large and the success probability is small, and it converges to the Normal distribution as λ increases. These findings are substantiated by theoretical derivations and supported with examples from current scientific applications. The study unifies theoretical derivations and practical illustrations, highlighting the central role of PD in statistical modeling and its versatility across various domains. The findings reinforce the importance of PD as a core tool in applied statistics and suggest potential for its enhanced application in complex systems through hybrid models and extensions involving fuzzy logic.
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