Solution
= 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 $\xi_i$: $E\sum_{i=1}^M\xi_i=\sum_iE[1_{M\ge i}]E\xi_i=\lambda^{-1}EM$.
= 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 $\xi_i$: $E\sum_{i=1}^M\xi_i=\sum_iE[1_{M\ge i}]E\xi_i=\lambda^{-1}EM$.