One mixed negative binomial distribution with application

In this paper we do some research on a three-parameter distribution which is called beta-negative binomial (BNB) distribution, a beta mixture of negative binomial (NB) distribution. The closed form and the factorial moment of the BNB distribution are derived. In addition, we present the recursion on the pdf of BNB stopped-sum distribution, and make stochastic comparison between BNB and NB distributions. Furthermore, we have shown that BNB distribution has heavier tail than NB distribution. The application of BNB distribution is carried out on one sample of insurance data. Based on the results, we have shown that the BNB provides a better fit compared to the Poisson and the NB for count data.