Binomial Mixtures Based on Beta I Distribution and Its Generalizations with Application to a Two Stage Group Screening Design

Abstract

When mixed, two distributions can form another distribution. Binomial distribution and beta distributions combine to form a binomial mixture with the latter being a prior distribution which is a continuous distribution. Skellam pioneered this study when he mixed binomial distribution with its parameter p taking beta distribution. The paper focused on construction of binomial mixtures, their properties, special cases and the application of the mixtures in a two stage group screening design. The two methods of constructing mixtures were proved to yield identical results. The binomial mixtures obtained were expressed in recursive forms and their corresponding differential equations obtained. The moments obtained indicated that binomial mixtures are probability density functions. Applied to two stage group screening design, binomial mixtures have demonstrated reduced number of tests which are more cost effective than individual testing.