A hospital keeps records of its emergency-room traffic and the number of patients arriving at the emergency-room. Those records indicate that, beginning at 6:00 pm on any given day, the number of patients arriving at the emergency-room has a Poisson process with parameter λ and that the expected time between patients is 1/λ, where time is measured in hours.
(a) Given that the first patient doesn't arrive by 6:15 p.m., determine the probability that he or she arrives by 6:45 p.m.
(b) Let X be the time until the third patient will arrive at the emergency-room. Determine the variance of 1/X.