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Probability Inequalities are useful for bounding quantities that might otherwise be hard to compute. A related concept is a concentration inequality, which specifically provides bounds on how far a random variable deviates from some value.

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Tight bound for Binomial distribution or, equivalently, the Incomplete Beta function?

Suppose $X \sim Binomial(n,p)$ with known $n$ but unknown $p$, and let $G(p,k) = P[X \geq k)$ for $k=0, \ldots, n$. I am looking for a tight upper bound on $|G(p_1, k) - G(p_2, k)|$ for some given $k$ …
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