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

2013; Society for Industrial and Applied Mathematics; Volume: 57; Issue: 3 Linguagem: Inglês

10.1137/s0040585x97986138

1095-7219

A. M. Zubkov, Alexander Serov,

Financial Risk and Volatility Modeling

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.