Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 208 / 4 / d / Solution

Part (a) only gives a standard-deviation scale of order $n$, and part (b) gives the same concentration scale. Part (c) improves this to order $\sqrt{n}$. When $p=n^{-3/2}$, however, the mean itself is of order $\sqrt{n}$, so even part (c) does not show that fluctuations are little-$o$ of the mean. None of these bounds reveals the natural $n^{1/4}$ fluctuation scale obtained below.

Solved by gpt-5.6-sol high.