Some approximations to the binomial distribution function

Abstract

Let p be given, 0 < p < 1. Let n and k be positive integers such that np ≤ k ≤ n, and let B_n(k) = ∑ⁿ_r=k(ⁿ _r)p^r q^n-r, where q = 1 - p. It is shown that B_n(k) = [(ⁿ _k) p^k,q^-k] qF(n + 1, 1; k + 1; p), where F is the hypergeometric function. This representation seems useful for numerical and theoretical investigations of small tail probabilities. The representation yields, in particular, the result that, with A_n(k) = [(ⁿ _k)p^k q^n-k+IJ [(k + 1)/(k + 1 - (n + l)p)], we have 1 ≤ A_n(k)/B_n(k) ≤ 1 + x^-2, where x = (k - np)/(npq)^t. Next, let N_n(k) denote the normal approximation to B_n(k), and let C_n(k) = (x + √q/np)√2π exp [x²/2]. It is shown that (A_nN_nC_n)/B_n → 1 as n → ∞, provided only that k varies with n so that x ≥ 0 for each n. It follows hence that A_n/B_n → 1 if and only if x → ∞ (i.e. B_n → 0). It also follows that N_n/B_n → 1 if and only if A_nC_n → 1. This last condition reduces to x = o(n^l/6) for certain values of p, but is weaker for other values; in particular, there are values of p for which N_n/ B_n can tend to one without even the requirement that k/n tend to p.