Given $x \in \left[0\,;\,1\right]$, is there a closed form solution (or a very good approximation?) for the tightest (i.e. minimal) offset $l$ such that : $$ \sum_{k\in\left[m - l\,;\,m+l\right]} {n \choose k}\,p^k\,(1-p)^{n-k} > x $$ where $$m = (n+1)\,p - 1$$ is the mode, that is the value of $k$ that maximizes the function $k \mapsto {n \choose k}\,p^k\,(1-p)^{n-k}$.
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Usually, these types of bounds do not have closed form solutions. – Michael Burr May 29 '18 at 19:12
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This is dubious. Even the cumulative distribution (i.e. as if the lower bound was $0$) doesn't have a closed-form, and you want to find its inverse. The variable lower bound makes is harder. And if you try with the Gaussian approximation, you'll get a continuum of solutions. – May 29 '18 at 19:16
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You changed the question since my previous comment. The problem has now a single unknown. The conclusion remains: closed-form highly improbable. – May 29 '18 at 19:18
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@YvesDaoust yes I edited for clarity after a first comment asking for the meaning of "tight" (this comment is now deleted). I'm really looking for something symmetric around the mode so a single offset is enough – Julien__ May 29 '18 at 19:22