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I've read in a paper that the following binomial distribution

[tex]\sum_{k=floor(N/2)+1}^N{N\choose k}\varepsilon^k(1-\varepsilon)^{N-k}[/tex]

can be upper bounded using Chernoff bound by

[tex]e^{ floor(N/2)}\,\Phi(s_0)[/tex]

where

[tex]\Phi(s)=\left(1-\varepsilon(1-e^s)\right)^N[/tex]

and

[tex]floor(N/2)\,\Phi(s_0)=\frac{\partial}{\partial s}\left. \Phi(s)\right|_{s=s_0}[/tex]

Could anyone explain to me how?

Thanks

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# Chernoff Bound for Binomial Distribution

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