Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 1 / b / Solution

For every $0<\lambda <1/\alpha$, the Chernoff bound and the sub-exponential moment-generating bound giveIf $0<t\le \nu /\alpha$, choose $\lambda =t/\nu$; at the endpoint, take a limit from below. This yieldsIf $t>\nu /\alpha$, let $\lambda \uparrow 1/\alpha$. Since $\nu /(2{\alpha }^2)<t/(2\alpha )$,and hence $\mathbb{P}(X\ge t)\le e^{-t/(2\alpha )}$.