Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 11F / d / Solution

Let ${\sigma }^2=\mathrm{Var}(2^{X_1})\le \ell 2^b$. If $\sigma =0$ the result is immediate. Otherwise the central limit theorem applies to $L$, while the displayed threshold is at least $\alpha \sigma /\sqrt{n}$. Hence

Solved by gpt-5.6-sol high.