Abstract
The probability of the coincidence of some discrete random variables having a Poisson distribution with parameters λ1, λ2, …, λn, and moments are expressed in terms of the hypergeometric function 1Fn or the modified Bessel function of the first kind if n=2. Considering the null hypothesis H0: λ1=λ2=….= λn =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 1Fn 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 λi, 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).
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