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Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$.

I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$.

Specifically, if $\delta,p=o(1)$ are not constants, it seems that $f(\delta)$ should satisfy $f(\delta)=\omega(np)$.

Can we find any $f(\delta)=\omega(np)$ such that $\Pr[X > f(\delta)] \ge \delta$?

For example, assume that $p=\delta=1/n$. Am I correct to assume that such $f=\omega(1)$ has to exist?

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  • $\begingroup$ "For example..." I don't think so, because the Binomial$(n, 1/n)$ distribution tends to the Poisson$(1)$ distribution as $n \to \infty$, so for a given $\delta$ the corresponding $f(\delta)$ should tend to a fixed constant. Right? $\endgroup$
    – usul
    Commented Jan 2, 2018 at 16:13
  • $\begingroup$ @usul - I'm not sure I follow. If $\delta$ is tiny (say $e^{-n}$), why can't we have $f(\delta)=\omega(1)$ for Poisson(1)? All that is required is that a tiny ($\delta$) fraction of the probability mass will exceed $f(\delta)$. For example, is it not true that $\Pr[X > \log n]\ge e^{-n}$ for $X\sim$ Poisson$(1)$? $\endgroup$
    – R B
    Commented Jan 2, 2018 at 17:04
  • $\begingroup$ Oh, right. I somehow missed that $\delta$ is also shrinking as $n$ grows. Sorry! $\endgroup$
    – usul
    Commented Jan 2, 2018 at 20:20
  • $\begingroup$ What is $\omega$? How is $\delta$ supposed to be related to $n,p$? Can you state the problem formally? $\endgroup$
    – Iosif Pinelis
    Commented Jan 4, 2018 at 22:00
  • $\begingroup$ If $p=\delta=1/n$ and $f=1/2$ (say), then $P(X>f)=1-P(X=0)=1-(1-1/n)^n>1-1/e>1/2\ge1/n=\delta$ for $n\ge2$. $\endgroup$
    – Iosif Pinelis
    Commented Jan 4, 2018 at 22:07

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