Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 210 / 1 / c / Solution

Fix $0\le \lambda <c^{-1}$. Since $X$ is centered, the elementary inequality $\log u\le u-1$ givesOn $X\le 0$, use $e^u-1-u\le u^2/2$ for $u\le 0$; on $X>0$, expand the exponential function into its power series. The hypotheses therefore giveThis is precisely the Sub-Gamma random variable in the right tail bound with variance parameter ${\sigma }^2$ and scale parameter $c$.