Consider a renewal process N t with i.i.d. interarrival times i . Say, a piece of equipment serves until it breaks down; upon failure

Consider a renewal process N_t with i.i.d. interarrival times τ_i. Say, a piece of equipment serves until it breaks down; upon failure it is instantly replaced, and the renewal process continues.

(a) Suppose we want to “inspect” the process, fixing a time t and observing the length of the interarrival time interval that contains the time t. Say, we observe the lifetime of the equipment that was working at time t. Argue that the length of the time interval mentioned does not have the same distribution as τ_i’s and on the average, this length is larger then E{τ_i}.

(b) Answer the same question regarding the interarrival time that follows time t; say, the lifetime of the equipment installed after epoch t.