Mathematics and Statistics Vol. 10(6), pp. 1218 - 1228
DOI: 10.13189/ms.2022.100608
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Binomial-Geometric Mixture and Its Applications

Hussein Eledum 1,*, Alaa R. El-Alosey 2
1 Department of Statistics, Faculty of Science, University of Tabuk, KSA
2 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt


A mixture distribution is a combination of two or more probability distributions; it can be obtained from different distribution families or the same distribution families with different parameters. The underlying distributions may be discrete or continuous, so the resulting mixture probability distribution function should be a mass or density function. In the last few years, there has been great interest in the problem of developing a mixture distribution based on the binomial distribution. This paper uses the probability generating function method to develop a new two-parameter discrete distribution called a binomial-geometric (BG) distribution, a mixture of binomial distribution with the number of trials (parameter ) taken after a geometric distribution. The quantile function, moments, moment generating function, Shannon entropy, order statistics, stress-strength reliability and simulating the random sample are some of the statistical highlights of the BG distribution that are explored. The model's parameters are estimated using the maximum likelihood method. To examine the performance of the accuracy of point estimates for BG distribution parameters, the Monte Carlo simulation is performed with different scenarios. Finally, the BG distribution is fitted to two real lifetime count data sets from the medical field. As a result, the proposed BG distribution is an overdispersed right-skewed and can accommodate a constant hazard rate function. The proposed BG distribution is appropriate for modelling the overdispersed right-skewed real-life count data sets and it can be an alternative to the negative binomial and geometric distributions.

Mixture of Distributions, Binomial Distribution, Geometric Distribution, Hazard Function, Maximum Likelihood Estimation, Quantile Functions, Shannon Entropy, Stress-strength Parameter

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Hussein Eledum , Alaa R. El-Alosey , "Binomial-Geometric Mixture and Its Applications," Mathematics and Statistics, Vol. 10, No. 6, pp. 1218 - 1228, 2022. DOI: 10.13189/ms.2022.100608.

(b). APA Format:
Hussein Eledum , Alaa R. El-Alosey (2022). Binomial-Geometric Mixture and Its Applications. Mathematics and Statistics, 10(6), 1218 - 1228. DOI: 10.13189/ms.2022.100608.