The probability of the coincidence of some discrete random variables having a Poisson distribution with parameters λ₁, λ₂, …, λ_n, and moments are expressed in terms of the hypergeometric function ₁F_n or the modified Bessel function of the first kind if n=2. Considering the null hypothesis H₀: λ₁=λ₂=….= λ_n =θ, where θ is some positive constant number, asymptotic approximations of the probability and moments are derived for large θ using the asymptotic expansion of the hypergeometric function ₁F_n and that of the modified Bessel function of the first kind if n=2. Further, we show that if the sample mean is a minimum variance unbiased estimator (MVUE) for the parameter λ_i, then the probability that H₀ is true can be approximated by that of a coincidence. In that case, a chi-square χ² goodness of fit test can be established and a 100(1-α)% confidence interval (CI) for θ can be constructed using the variance of the coincidence (or via coincidence) and the Central Limit Theorem (CLT).

probability, hypergeometric functions, modified bessel functions, asymptotic expansion, MVUE