If $X$ is a sum of i.i.d. random variables taking values in $\{0,1\}$ and $E[X]=\mu$, the Chernoff bound tells us that$$\Pr(X\geq (1+\delta)\mu)\leq e^{-\frac{\delta^2\mu}{3}}$$for all $0<\delta<1$.If $E[X]\leq\mu$ instead, does there exist a constant $c$ such that $$\Pr(X\geq (1+\delta)\mu)\leq e^{-c\delta^2\mu}$$for all $0<\delta<1$?(This question is related, but the answer there doesn't help.)
Chernoff bound when we only have upper bound of expectation
probability theory;chernoff bounds