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

Abstract

Let the following expressions denote the binomial and Poisson probabilities,\begin{equation*}\begin{align*}\tag{1.1}B(k; n, p) &= \sum^k_{j=0}b(j; n, p) \\ &= \sum^k_{j=0} \binom{n}{j}p^j (1 - p)^{n-j}, \\ \tag{1.2}P(k;λ) &= \sum^k_{j=0}p(k; λ) = \sum^k_{j=0} e^{-λ}λ^k/k\end{align*}!.\end{equation*}Section 2 contains two basic theorems which generalize results ofAnderson and Samuels [1] and Jogdeo [7]. These two theorems serveas lemmas for the more detailed results of Sections 3 and 4. Section3 is devoted to a study of the median number of successes in Poissontrials (i.e. independent trials where the success probability mayvary from trial to trial). The study utilizes a method first introducedby Tchebychev [12], generalized by Hoeffding [6], and used by Darroch[5] and Samuels [10]. The results correspond to those for the modalnumber of successes obtained by Darroch. Ramanujan (see [8]) consideredthe following equation, where n is a positive integer: \begin{equation*}\tag{1.3}\frac{1}{2}= P(n - 1; n) + y_n p(n; n),\end{equation*} and correctly conjecturedthat \frac{1}{3} < y_n < z_{k,n} < \frac{2}{3} and, for each k and for n ≥ 2k, zk,n decreasesto yk as n → ∞.