Mathematical Properties of the Binomial-Poisson Distribution

This study conducts a thorough analytical exploration of the Binomial-Poisson distribution, a compound probability model where the number of Binomial trials follows a Poisson distribution. Through application of the compound distribution methodology and law of total probability, we derive its probability mass function (PMF) and demonstrate its simplification to a Poisson distribution with parameter. Key properties, including the cumulative distribution function (CDF), hazard function, moment generating function (MGF), probability generating function (PGF), and moments (mean, variance, Skewness, and kurtosis), are systematically established. Maximum likelihood estimation (MLE) is applied for parameter inference, and a comparative analysis with the Binomial and Poisson distributions highlights its distinct tendency to concentrate around smaller values of the random variable. The findings confirm the Binomial-Poisson distribution as a flexible and robust model for discrete event patterns, offering valuable applications in stochastic frequency analysis.