 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 ${\xi }_i$: $E{\sum }_{i=1}^M{\xi }_i={\sum }_{i}E[1_{M\ge i}]E{\xi }_i={\lambda }^{-1}EM$.