Derivation of Poisson Xrama Distribution and Its Properties

This study introduces the Poisson Xrama Distribution, a model for analyzing count data that exhibits overdispersion. By combining the Poisson distribution with the Xrama distribution, this model addresses the limitations of traditional Poisson models, which assume equidispersion. The Poisson Xrama Distribution offers enhanced flexibility in handling variance inflation, making it suitable for scenarios where standard Poisson models are insufficient. Key statistical properties, including moments, variance, skewness, kurtosis and index of dispersion measures are derived. Maximum likelihood estimation is employed for parameter estimation, providing a robust framework for practical applications. This distribution is particularly useful in fields where count data often display overdispersion, such as biology and economics, offering a promising alternative to existing distribution models.