Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 210 / 1 / b / Solution

The Hoeffding lemma for a Bernoulli distribution variable gives $\mathbb{E}{e}^{s(B-p)}\le e^{s^2/8}$. Multiplication of the moment-generating functions of the independent random variables gives $\mathbb{E}{e}^{s(X-np)}\le e^{n{s}^2/8}$. Thus the centered count is a sub-Gaussian random variable with variance proxy $n/4$. The Chernoff bound, optimized at $s=4t/n$, givesThe same estimate holds for the lower tail by applying the Chernoff bound to $-(X-np)$. No independence between different bin counts is needed.