Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 1 / b / Solution

SetUnder the probability measure with density $Z$, the Jensen inequality for the concave logarithm givesThe sub-Gaussian assumption with variance parameter $\nu /4$ gives $M(2\lambda )\le e^{\nu {\lambda }^2/2}$. A second application of Jensen inequality gives $M(\lambda )\ge e^{\lambda \mathbb{E}X}=1$. Consequentlyas required.