Some Approximations to the Binomial Distribution Function

Let p be given, 0 < p < 1. Let n and k be positive integers such that np k n, and let Bₙ (k) = ⁿₑ=₊ nr pʳq^n-r, where q = 1 - p. It is shown that Bₙ (k) = k pᵏq^n - 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) = kpᵏq^n - k + 1 (k + 1) / (k + 1 - (n + 1) p), we have 1 Aₙ (k) /Bₙ (k) 1 + x^-2, where x = (k - np) / (npq) ^1{2}. Next, let Nₙ (k) denote the normal approximation to Bₙ (k), and let Cₙ (k) = (x + q/np) 2 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 NₙNₙ 1 if and only if AₙCₙ 1. This last condition reduces to x = o (n^1/6) 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.