Binomial Mixture Based on Generalized Four Parameter Beta Distribution as Prior

A probability distribution can be constructed by mixing two distributions. Binomial distribution when compounded with beta distribution as prior forms a binomial mixture that is a continuous distribution. Skellam 1948, mixed a binomial distribution with its parameter being the probability of success considered as a random variable taking beta distribution. Probability distributions with binomial outcome tend to fail to fit empirical data due to over-dispersion. To address this challenge binomial mixtures are modeled to cater for the influence caused by over-dispersion. This paper focuses on binomial mixture with a four parameter generalized beta mixing distributions. In particular it focuses on application of McDonald generalized and Gerstenkon generalized mixing distributions. The binomial mixture obtained is proved to be a probability density function. Its moments are obtained using probability generating function techniques. The binomial mixture obtained can be applicable to probability distributions whose outcome are binomial in nature.