Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 224 / 4 / d / Solution

For the uniform distribution, the code in part c hasbecause every summand is at most ${\log }_{2}m$ and at least one is strictly smaller. For $m=3$, the explicit code $C(1)=\lambda$, $C(2)=0$, $C(3)=1$ has mean length $2/3<{\log }_{2}3$.