Let N(t) be the number of failures of a computer system in the interval [0,t]. We suppose that {N(t),t ≥ 0} is a Poisson process with rate λ = 1 per week.
(a) Calculate the probability that
(i) the system operates without failure during two consecutive weeks,
(ii) the system will have exactly two failures during a given week, knowing that it operated without failure during the previous two weeks,
(iii) less than two weeks elapse before the third failure occurs.
(b) Let
