Unit Probability Distributions: A Comprehensive Review of Models, Properties, and Applications

Unit probability distributions defined on the standard interval (0, 1) serve as foundational tools for modeling data constrained within this bounded domain. Such data frequently emerge in disciplines where proportions, rates, and probabilities are analyzed, including economics, finance, hydrology, environmental sciences, biomedical research, and reliability engineering. Traditional models such as the Beta and Kumaraswamy distributions have long provided flexible frameworks for these applications. However, the increasing complexity of real-world phenomena has spurred the development of more versatile and specialized unit distributions. This article presents a comprehensive review of the literature on unit probability distributions, encompassing both classical models and recent innovations. Emphasis is placed on transformation techniques used to generate new families, key analytical properties, and a comparative evaluation of estimation methods. A diverse array of real-world applications is examined, highlighting the practical relevance and empirical performance of modern unit distributions across multiple domains. By synthesizing these developments, the review offers a structured resource to support further methodological advancement and informed model selection for bounded data analysis.