A specific component in a cryptometer has an Exp(µ)-distributed life- time, µ > 0. If replacement is made as soon as a component fails, and if X(t) = # failures during (0, t] = # replacements during (0, t], then {X(t), t ≥ 0} is, of course, a Poisson process. Let {Vn, n ≥ 1} be these usual interreplacement times, and suppose, instead, that the nth component is replaced:
(a) After time min{Vn, a}, that is, as soon as the component fails or reaches age a, whichever comes first. Show that the replacement process is not a Poisson process.
(b) After time min{Vn, Wn}, where {Wn, n ≥ 1} is a sequence of independent, Exp(θ)-distributed random variables, θ > 0, which is indepen- dent of {Vn, n ≥ 1}. Show that the replacement process is a Poisson process and determine the intensity.