A Modified Binomial Distribution Approximation by Poisson Distribution

This paper examines the generalized Binomial distribution introduced by Dwass (1979), a flexible discrete probability model that extends the classical Binomial distribution. The model also encompasses other important distributions such as the hypergeometric and Pólya distributions as special cases. It is defined through four parameters AAA, BBB, nnn, and ααα, where AAA and BBB are positive real numbers, nnn is a positive integer, and ααα is a real-valued parameter subject to the constraint (n−1)α≤A+B(n-1)α ≤ A + B(n−1)α≤A+B. The distribution is characterized by factorial-type expressions A(k)A⁽ᵏ⁾A(k) and B(n−k)B⁽ⁿ⁻ᵏ⁾B(n−k), providing a broad framework for modeling dependent and over-dispersed count data. The probability mass function, along with its mean and variance, demonstrates the distribution’s flexibility in capturing a wide range of stochastic behaviors. In particular, the mean is given by μ=AnA+Bμ = An/A+Bμ=A+BAn, while the variance incorporates the effect of the additional parameter ααα, allowing for adjustment of dispersion beyond the classical Binomial case. This study highlights the usefulness of the generalized Binomial distribution in applied probability and statistical modeling