An inequality involving multinomial probabilities

Abstract

If F 1 , …, F k are multunomially distributed, then for any a 1 , …, a k , P ( F 1 ≤ a 1, …, F k ≤ a k ) ≤ P ( F 1 ≤ a 1 ) … P ( F k ≤ a k ) For Some values of the parameters this simple upper bound is sharper than that provided by the second Bonferroni inequality. A similar result holds for the probability that F 1 , …, F k all exceed certain limits.