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We know from Chernoff bound $P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where $X$ follows Binomial($N, \frac{1}{2}$). If I take $N=1000, \epsilon=0.01$, the upper bound is 0.82. However, the actual value is 0.27. Can we improve this Chernoff bound?
In probability theory, the Chernoff bound, named after Herman Chernoff but due to Herman Rubin, gives exponentially decreasing bounds on tail distributions of sums of independent random variables. It is a sharper bound than the known first- or second-moment-based tail bounds such as Markov's inequality or Chebyshev's inequality, which only yield power-law bounds on tail decay. However, the Chernoff bound requires that the variates be independent – a condition that neither Markov's inequality nor Chebyshev's inequality require, although Chebyshev's inequality does require the variates to be pairwise...
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or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent? The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and Perlmutter paper, "Generalizations of the Kunen inconsistency", Annals of Pure and Applied Logic, 163 (201...
I'd guess that if the tag (proof-analysis) is supposed to be used in some consistent manner, it would be good to have some guidance in the tag-info mathoverflow.net/tags/proof-analysis/info - which is currently empty.
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