Monotone Convergence of Binomial Probabilities and a Generalization of Ramanujan'S Equation

摘要

The paper contains two basic theorems which generalize results of Anderson and Samuels (1967) and Jogdeo (1967). These two theorems serve as lemmas for the more detailed results of Sections 3 and 4. Section 3 is devoted to a study of the median number of successes in Poisson trials (i.e. independent trials where the success probability may vary from trial to trial). The study utilizes a method first introduced by Tchebychev (1846), generalized by Hoeffding (1956) and used by Darroch (1964) and Samuels (1965). The results correspond to those for the modal number of successes obtained by Darroch (1964). (Author)