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

Let the following expressions denote the binomial and Poisson probabilities, {equation*}{align*}{1.1}B(k; n, p) &= ∑^kⱼ₌₀ b(j; n, p) \\ &= ∑^kⱼ₌₀ {n}{j}p^j (1 - p)ⁿ⁻ʲ, \\ {1.2}P(k; λ) &= ∑^kⱼ₌₀p(k; λ) = ∑^kⱼ₌₀ e-λλ^k/k{align*}!.{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*}{1.3}1/2 = P(n - 1; n) + y_n p(n; n),{equation*} and correctly conjectured that 1/3 < yₙ < 1/2. In Section 4 we show that for the corresponding binomial equation, {equation*}{1.4}1/2 = B(k - 1; n, k/n) + zk,nb(k; n, k/n),{equation*} 1/3 < zk,n < 2/3 and, for each k and for n 2k, zk,n decreases to yₖ as n → ∞.