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?
Practical forms of Chernoff bound for inequality in expectation
probability theory;chernoff bounds
It is more practical just because it has the advantage of being easier to state and compute with in many situations (quoted from [1]).I am not sure whether (i) still holds for $\mu' = E[X] \le \mu$. At least, we cannot obtain that directly from the following computation:$$Pr[X \ge (1 + \delta) \mu] \le Pr[X \ge (1 + \delta) \mu'] \le e^{-\frac{\delta^2 \mu'}{3}} \ge e^{-\frac{\delta^2 \mu}{3}}$$[1] Mitzenmacher, Michael and Upfal, Eli (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press. ISBN 0-521-83540-2.