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

Let the following expressions denote the binomial and Poisson probabilities, equation*align*1. 1B (k; n, p) n, p) \\) ) = ᵏ₉=₀ e^-ᵏ/kalign*!. equation* Section 2 contains two basic theorems which generalize results of Anderson and Samuels 1 and Jogdeo 7. 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 12, generalized by Hoeffding 6, and used by Darroch 5 and Samuels 10. The results correspond to those for the modal number of successes obtained by Darroch. Ramanujan (see 8) considered the following equation, where n is a positive integer: equation*1. 312 = P (n - 1; n) + yₙ p (n; n), equation* and correctly conjectured that 13 < yₙ < 12. In Section 4 we show that for the corresponding binomial equation, equation*1. 412 = B (k - 1; n, k/n) + z₊, ₍b (k; n, k/n), equation* 13 < z₊, ₍ < 23 and, for each k and for n 2k, z₊, ₍ decreases to yₖ as n.