I see the following variation of Chernoff Bound in one of the papers of a very good conference. The Chernoff Bound is given in the paper is as follows:
Let $X_1,..., X_T$ be independent random variables with $E[X_i] = p_i$ and $X_i \in [0, 1]$. Let $X = \sum_{i=1}^T X_i$ and $\mu = \sum_{i=1}^T p_i = E[X]$. Then, for all $\delta\geq 0$
- $P[X \geq (1 + \delta)\mu] \leq e^{-\frac{\delta^2}{2+\delta}\mu}$ (1)
- $P[X \leq (1-\delta)\mu] \leq e^{-\frac{\delta^2}{2+\delta}\mu}$ (2)
I do not have a problem with (1) but I have not seen (2) before. Is this one of the variations of the Chernoff bound?