1) At the beginning of the term, students sign up for advising appointments at a Poisson rate of one student every 30 minutes. If 70% of the appointments are in-person, and the rest are virtual, (a) (5 points) What is the probability that at least one in-person appointment is requested during a 3-hour period? (b) (5 points) Given that 7 in-person appointments are requested in 2 hours, what is the expected number of virtual appointments requested in that time? (c) (5 points) Given that 7 in-person appointments are requested in 2 hours, what is the expected total number of appointments requested in that time? (d) (10 points) If 5 appointments are requested in an hour, what is the probability that exactly 2 of them are in-person appointments?

2) Let {N(t),t 0} be a Poisson process with rate . Let Sn denote the time of the nth event.