Ordering comparison of negative binomial random variables with their mixtures

In this paper we compare a negative binomial distribution having a fixed probability of success with a negative binomial distribution having a random probability of success. Comparisons are made with respect to various types of orderings, such as, simple stochastic ordering, likelihood ratio ordering, uniformly more variable ordering, convex ordering and expectation ordering. As a special case, the two distributions are compared when they have the same mean.