Some inequalities among binomial and Poisson probabilities

k (1.4) P(k; X) = E p(j; X), k = 0, 1, i-o for X = np. In this paper it is shown that the error of approximation of the binomial cumulative distribution function P(k; np) B(k; n, p) is positive if k < np np/(n + 1) and is negative if np < k. In fact, B(k; n, X/n) is monotonically increasing for all n (2 X) if k < X 1 and for all n > k/(X k) if X 1 < k < X, and is monotonically decreasing for all n (2 k) if X < k. Thus