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

A leaf at depth $d$ is reached with probability $2^{-d}$ and contributes $2^d$. Summing over leaves gives $\mathbb{E}[2^{X_1}]=\ell$. Since $L$ is the sample mean of the $2^{X_i}$, $\mathbb{E}L=\ell$.

Solved by gpt-5.6-sol high.