Some Approximations to the Binomial Distribution Function
Let p be given, 0 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+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.