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June, 1977 Distribution Inequalities for the Binomial Law
Eric V. Slud
Ann. Probab. 5(3): 404-412 (June, 1977). DOI: 10.1214/aop/1176995801


We prove that the probability of at least $k$ successes, in $n$ Bernoulli trials with success-probability $p$, is larger than its normal approximant if $p \leqq \frac{1}{4}$ and $k \geqq np$ or if $p \leqq \frac{1}{2}$ and $np \leqq k \leqq n(1 - p)$. A local refinement is given for $np \leqq k \leqq n(1 - p), k \geqq 2$, and for $p \leqq \frac{1}{4}, k \geqq n(1 - p)$. Bounds below for individual binomial probabilities $b(k, n, p)$ are also given under various conditions. Finally, we discuss applications to significance tests in one-way layouts.


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Eric V. Slud. "Distribution Inequalities for the Binomial Law." Ann. Probab. 5 (3) 404 - 412, June, 1977.


Published: June, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0358.60015
MathSciNet: MR438420
Digital Object Identifier: 10.1214/aop/1176995801

Primary: 60C05
Secondary: 62E15

Keywords: binomial , conservative test , Poisson and normal laws , tail probabilities

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 3 • June, 1977
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