A Complete Proof of Universal Inequalities for the Distribution Function of the Binomial Law

We present a new form and a short complete proof of explicit two-sided estimates for the distribution function $F_{n,p}(k)$ of the binomial law with parameters $n,p$ from [D. Alfers and H. Dinges, Z. Wahrsch. Verw. Geb., 65 (1984), pp. 399--420]. These inequalities are universal (valid for all values of parameters and argument) and exact (namely, the upper bound for $F_{n,p}(k)$ is the lower bound for $F_{n,p}(k+1)$). Such estimates allow to bound any quantile of the binomial law by two subsequent integers that it contains.