Question 7.EX.8: A machine in use is replaced by a new machine either when it......

(a) The number of replaced machines by time t constitutes a renewal process. The time between replacements equals T, if lifetime of new machine is $\geqslant T ; x$, if lifetime of new machine is x, x<T. Hence,
$E \text { [time between replacements }]=\int_0^T x f(x) d x+T[1-F(T)]$
and the result follows by Proposition 3.1.
(b) The number of machines that have failed in use by time t constitutes a renewal process. The mean time between in-use failures, E[F], can be calculated by conditioning on the lifetime of the initial machine as E[F]= E[E[F $\mid$ lifetime of initial machine ]]. Now
$E[F \mid \text { lifetime of machine is } x]= \begin{cases}x, & \text { if } x \leqslant T \\ T+E[F], & \text { if } x>T\end{cases}$
Hence,
$E[F]=\int\limits_0^T x f(x) d x+(T+E[F])[1-F(T)]$
or
$E[F]=\frac{\int_0^T x f(x) d x+T[1-F(T)]}{F(T)}$
and the result follows from Proposition 3.1.