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Skellam distribution, Wikipedia, The free encyclopedia, 2016. [Online] Available: http://en.wikipedia.org/wiki/Skellam_distribution. [June 15, 2016]

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## Article

# Coincidences, Goodness of Fit Test and Confidence Interval for Poisson Distribution Parameter via Coincidence

^{1}School of Mathematics and Statistics, Carleton University, Ottawa, Canada

*American Journal of Applied Mathematics and Statistics*.

**2016**, Vol. 4 No. 6, 185-193

**DOI:**10.12691/ajams-4-6-4

**Copyright © 2017 Science and Education Publishing**

**Cite this paper:**

Victor Nijimbere. Coincidences, Goodness of Fit Test and Confidence Interval for Poisson Distribution Parameter via Coincidence.

*American Journal of Applied Mathematics and Statistics*. 2016; 4(6):185-193. doi: 10.12691/ajams-4-6-4.

Correspondence to: Victor Nijimbere, School of Mathematics and Statistics, Carleton University, Ottawa, Canada. Email: victornijimbere@gmail.com

## Abstract

The probability of the coincidence of some discrete random variables having a Poisson distribution with parameters

*λ*_{1}*,**λ*_{2}*,**…, λ*_{n}, and moments are expressed in terms of the hypergeometric function_{1}*F*_{n}or the modified Bessel function of the first kind if*n=2*. Considering the null hypothesis*H*_{0}*: λ*_{1}*=λ*_{2}*=….= 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_{1}*F*_{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*H*_{0}is true can be approximated by that of a coincidence. In that case, a chi-square*χ*^{2 }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).