“Practical forms” of Chernoff bound for inequality in expectation
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02-11-2019 - |
質問
From Wikipedia:
The above formula is often unwieldy in practice, so the following looser but more convenient bounds are often used:
(i) $Pr(X\geq (1+\delta)\mu)\leq e^{-\frac{\delta^2\mu}{3}}, 0<\delta<1$
(ii) $Pr(X\leq (1-\delta)\mu)\leq e^{-\frac{\delta^2\mu}{2}}, 0<\delta<1$
The assumption they use is $E[X]=\mu$.
Would (i) still hold if we only assume $E[X]\leq \mu$? Would (ii) still hold if we only assume $E[X]\geq\mu$?
If not, what "practical forms" do we have in these cases?
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