Telephone calls to an emergency number arrive according to a nonhomogeneous Poisson process whose intensity function is given by
where t is in hours, and λ{t) = λ{t - 24), for t ≥ 24. Furthermore, the duration (in minutes) of a call received at night has a uniform distribution on the interval (0,2], whereas the duration of a call received during the day has
a uniform distribution on the interval (0,3]. Finally, the durations of calls are independent random variables.
(a) Calculate the probability that an arbitrary telephone call received at night will be longer than a given call received during the day.
(b) Calculate the variance of the number of calls received in the course of a given week.
(c) Let D be the total duration of the calls received during a given day. Calculate the variance of D.