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; \lambda) &= \sum^k_{j=0}p(k; \lambda) = \sum^k_{j=0} e^{-\lambda}\lambda^k/k\end{align*}!.\end{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: \begin{equation*}\tag{1.3}\frac{1}{2} = P(n - 1; n) + y_n p(n; n),\end{equation*} and correctly conjectured that $\frac{1}{3} < y_n < \frac{1}{2}$. In Section 4 we show that for the corresponding binomial equation, \begin{equation*}\tag{1.4}\frac{1}{2} = B(k - 1; n, k/n) + z_{k,n}b(k; n, k/n),\end{equation*} $\frac{1}{3} < z_{k,n} < \frac{2}{3}$ and, for each $k$ and for $n \geqq 2k, z_{k,n}$ decreases to $y_k$ as $n \rightarrow \infty$.