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) = ∑ⁿᵣ₌ₖ nr pʳqⁿ⁻ʳ, where $q = 1 - p$. It is shown that Bₙ(k) = nk pᵏqn - 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) = nkpᵏqn - 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) √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 NₙNₙ → 1 if and only if AₙCₙ → 1. This last condition reduces to x = o(n1/6) 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.