Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 11F (Probability)

For a nonnegative random variable $Z$ and $a>0$, the pointwise inequality $Z\ge a{1}_{\{Z\ge a\}}$ gives

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 $2^{X_i}$, $\mathbb{E}L=\ell$.

Similarly,Thus $\mathrm{Var}(L)\le \ell 2^b/n$. Chebyshev's inequality gives

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