A dedicated professor has been holding infinitely long office hours. Undergraduate students arrive according to a Poisson process at a rate of per hour, while graduate students arrive according to a second, independent Poisson process at a rate of per hour. An arriving student receives immediate attention (the previous student's stay is immediately terminated) and stays with the professor until the next student arrives. (Thus, the professor is always busy, meeting with the most recently arrived student.)

Question

(1) What is the probability that exactly three undergraduates arrive between 10:00 pm and 10:30 pm?

- unanswered

(2) What is the expected length of time in hours that the 10th arriving student (undergraduate or graduate) will stay with the professor?

- unanswered

(3) Given that the professor is currently talking with an undergraduate, what is the expected number of subsequent student arrivals up to and including the next graduate student arrival?

- unanswered

(4) Given that the professor is currently talking with an undergraduate, what is the probability that 5 of the next 7 students to arrive will be undergraduates?

- unanswered

As rumors spread around campus, a worried department head drops in at midnight and begins observing the professor.
 
(5) Beginning at midnight, what is the expected length of time until the next student arrives, conditioned on the event that the next student will be an undergraduate?

- unanswered

(6) What is the expected time that the department head will have to wait until the set of students he/she has observed meeting with the professor (including the student who was meeting the professor when the deparment head arrived) include both an undergraduate and a graduate student?

Answer 1

1. 0.125
2. 1/8
3. ???
4. 0.0608
5. 1/8
6. ???

Answer 2

3. 1.6
6.0.283333