"Practical forms" of Chernoff bound for inequality in expectation From Wikipedia:

The above formula is often unwieldy in practice, so the following looser but more convenient bounds are often used:
(i) $Pr(X\geq (1+\delta)\mu)\leq e^{-\frac{\delta^2\mu}{3}}, 0&lt;\delta&lt;1$
(ii) $Pr(X\leq (1-\delta)\mu)\leq e^{-\frac{\delta^2\mu}{2}}, 0&lt;\delta&lt;1$

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 &quot;practical forms&quot; do we have in these cases?
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.