From [Wikipedia](http://en.wikipedia.org/wiki/Chernoff_bound#Theorem_for_multiplicative_form_of_Chernoff_bound_.28relative_error.29): > 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?