Two Estimates of the Binomial Distribution

The tail of the binomial distribution is defined as {equation*}{1}E(n, s, p) = ∑^nᵣ₌ₛ {n}{r} p^r (1 - p)ⁿ⁻ʳ{equation*} with the restriction that p be non-negative and less than (approximately) $s/n$. An estimate of $E(n, s, p)$ can be considered as the product of two separate estimates, an estimate of the size of the leading term in (1) and an estimate of the ratio of $E(n, s, p)$ to the leading term. This paper is concerned solely with the second estimate. That is, let {equation*}{2}R = E(n, s, p) / {n}{s} p^s(1 - p)ⁿ⁻ˢ;{equation*} two estimates of R are presented. In Section 2 an upper bound to R is derived from a geometric series approach. The resulting bound is an improvement over previous results in that it is useful even when p is near $s/n$, as opposed to simpler geometric bounds, such as that of Bahadur [1], which either blow up or become excessively large when p is near $s/n$. The bound is given in Theorem 1, Equation (9). Section 3 discusses the error in using this bound. Section 4 gives a normal approximation to R, namely R^ of Equation (18). Theorem 3 shows that the relative error of this estimate goes to zero as s and $n - s$ go to infinity provided 0 p (s - 1)/(n - 1) and under a weak restriction on $s/n$ in the limit. The uniformity of this result with p contrasts with the many normal approximations to $E(n, s, p)$, such as those of Bernstein, [2] Feller, [6], [7] Uspensky, [9] and Camp [3] in which the relative error becomes infinite as p approaches zero. Section 5 presents a brief discussion of the behavior of R^.