Some approximations to the binomial distribution function

Bahadur, R. R. (1960) Some approximations to the binomial distribution function Annals of Mathematical Statistics, 31 (1). pp. 43-54. ISSN 0003-4851

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Let p be given, 0 < p < 1. Let n and k be positive integers such that np ≤ k ≤ n, and let Bn(k) = ∑nr=k(n r)pr qn-r, where q = 1 - p. It is shown that Bn(k) = [(n k) pk,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 An(k) = [(n k)pk qn-k+IJ [(k + 1)/(k + 1 - (n + l)p)], we have 1 ≤ An(k)/Bn(k) ≤ 1 + x-2, where x = (k - np)/(npq)t. Next, let Nn(k) denote the normal approximation to Bn(k), and let Cn(k) = (x + √q/np)√2π exp [x2/2]. It is shown that (AnNnCn)/Bn → 1 as n → ∞, provided only that k varies with n so that x ≥ 0 for each n. It follows hence that An/Bn → 1 if and only if x → ∞ (i.e. Bn → 0). It also follows that Nn/Bn → 1 if and only if AnCn → 1. This last condition reduces to x = o(nl/6) for certain values of p, but is weaker for other values; in particular, there are values of p for which Nn/ Bn can tend to one without even the requirement that k/n tend to p.

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