Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 210 / 1 / Solution

For $\lambda \in \mathbb{R}$, convexity of $u\mapsto e^{\lambda u}$ on $[0,1]$ givesConsequentlyFor $\lambda \ge 0$, independence and concavity of $u\mapsto \log \{1+u(e^{\lambda }-1)\}$ yieldThe Chernoff bound therefore gives, for $a=\bar{\mu }+x$,For $\bar{\mu }<a<1$, the minimizer satisfiesSubstitution gives the binary relative entropyThe endpoint cases follow by continuity.

Put $a=(1+\delta )\bar{\mu }$. The same calculation, followed by $\log (1-z)\le -z$, givesThusThe supplied lower bound $\log (1+\delta )\ge 2\delta /(2+\delta )$ implieswhich proves the second upper-tail estimate. If $(1+\delta )\bar{\mu }>1$, the event is empty and the same bound remains true.

For the lower tail, apply the exponential-moment argument with $\lambda <0$, or equivalently optimize at $a=(1-\delta )\bar{\mu }$. It givesFinally,for $0\le \delta <1$, yielding $e^{-n{\delta }^2\bar{\mu }/2}$; the case $\delta =1$ follows by a limit.