Bahadur, R. R. (1960) Some approximations to the binomial distribution function Annals of Mathematical Statistics, 31 (1). pp. 4354. ISSN 00034851

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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^{nr}, 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^{nk+I}J [(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 [x^{2}/2]. It is shown that (A_{n}N_{n}C_{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_{n}C_{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.
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