A Novel Probability Distribution: Mathematical Derivation and Validation of the Poisson Hamza Model

Abstract

This study introduces the Poisson Hamza Distribution (PHD), a novel probability distribution developed from the classical Poisson framework to address limitations in modeling count data. While the Poisson distribution is a standard tool for modeling rare events, its inherent assumptions, particularly equidispersion, limit its applicability in complex, real-world contexts. The PHD introduces enhanced modeling flexibility by accommodating overdispersion, thereby extending the utility of Poisson-based models. A comprehensive mathematical formulation of the PHD is presented, along with derivations of its key statistical properties, including moments, variance, standard deviation, skewness, and kurtosis. Theoretical validation is supported by empirical analysis, demonstrating the distribution’s robustness and practical relevance. These contributions offer a valuable extension to existing statistical methodologies and provide researchers and practitioners with an alternative model for analyzing overdispersed count data.