The Probability Distribution for the Number of Successes in Independent Trials

The number of successes S in n independent trials is one of the classic distributions in statistics. When the probability of success p is constant, S has the binomial distribution with parameters n and p. Distributional properties of S in the homogeneous case are well known, but they are considerably harder to establish when p varies. A brief historical review is undertaken to highlight some of the more important results obtained over the past 50 years. In spite of the progress made, there are still some important open problems concerning stochastic comparisons for S that deserve to be solved!