These two well-known Chernoff bounds for the sum of RVs $X=\sum_{k=1}^{n}X_k$ in mulitplicative form,
$\mathbf{P}(X \leq (1- \delta)\mathbf{E}X) \leq e^{-\frac{\delta^2 \mathbf{E}X}{2}}\\ \mathbf{P}(X \geq (1+ \delta)\mathbf{E}X) \leq \big(\frac{e^{\delta}}{(1+\delta)^{(1+\delta)}} \big)^{\mu} $
apply in the case when $X_k$ take values 0 and 1. I'm wondering if and how these results can extend in a more general case, e.g. when $X_k$ are exponentially distributed iid.