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