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Let $X_1,...,X_n$ be independent random variables, where $X_i=1$ with probability $p_i$ and $X_i=0$ with probability $1-p_i$. Let $X= \sum_{i=1}^n X_i$ and $\mu = E[X] = \sum_{i=1}^n p_i$. Then for $\alpha \in [0,1]$,

$\mathcal{P}[|X-\mu| \geq \alpha \mu] \leq 2\exp{\left(-\frac{\mu \alpha^2}{2}\right)}$

Relative entropy Chernoff bound

A tighter version of the standard Chernoff bound, which uses the Relative entropy