Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 33 / 5 / a / i / Solution

For a nonnegative random variable, $T={\int }_0^{\infty }1\{T>t\}dt$. Tonelli theorem permits interchanging expectation and this nonnegative integral, giving the tail integral formula for moments:The equality also holds when the mean is infinite. Thus mean completion time is the total area under the survival function, while a restricted mean survival time integrates only to a specified follow-up horizon.