*8. A machine in use is replaced by a new machine either when it fails or when it reaches the age of Ô years. If the lifetimes of successive machines are independent with a common distribution F having density / , show that

(a) the long-run rate at which machines are replaced equals

ô ç -é

(b) the long-run rate at which machines in use fail equals F(T)
\T 0xf(x)dx+ Til -F(T)]