Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 27K / a / Solution

This is Wald's identity. Expand the stopped sum and use $\{M\ge i\}$, which depends only on earlier interarrivals and is independent of ${\xi }_i$: $E{\sum }_{i=1}^M{\xi }_i={\sum }_{i}E[1_{M\ge i}]E{\xi }_i={\lambda }^{-1}EM$.

Solved by gpt-5.6-sol high.