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.)

(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?

1. 0.125

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

3. 1.6

6.0.283333

To solve these questions, we need to use the concepts of Poisson processes and conditional probability. Let's go through each question step by step and explain how to get the answers.

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

To calculate this probability, we can use the Poisson distribution. The Poisson distribution describes the number of events that occur in a fixed interval of time or space, given the average rate of occurrence. In this case, we have a Poisson process with an arrival rate of λ undergraduate students per hour.

The probability of exactly k events occurring in a Poisson process with rate λ is given by the formula:

P(X = k) = (e^(-λ) * λ^k) / k!

So, to find the probability that exactly three undergraduates arrive between 10:00 pm and 10:30 pm, we need to know the arrival rate λ for undergraduate students.

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

To calculate the expected length of time, we need to know the arrival rates for both undergraduate and graduate students. The expected length of time can be computed using the concept of waiting times in a Poisson process. We can use the fact that the waiting time for the kth arrival in a Poisson process with rate λ is equal to the gamma distribution with parameters k and 1/λ.

We can then calculate the expected waiting time for the 10th arriving student using the formula:

E(T) = k / λ

Where E(T) is the expected waiting time, k is the kth arrival (in this case, k = 10), and λ is the arrival rate.

(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?

To calculate the expected number of subsequent student arrivals, we need to use the concept of conditional probability. We want to find the expected number of students arriving until the next graduate student arrives, given that the professor is currently talking with an undergraduate.

We can use the fact that the number of arrivals in a Poisson process is memoryless and follows a geometric distribution. The expected number of arrivals until the first success (in this case, the next graduate student arrival) in a geometric distribution with parameter p is given by the formula:

E(N) = 1 / p

Where E(N) is the expected number of arrivals, and p is the probability of success in each arrival.

(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?

To calculate this probability, we need to use the concept of conditional probability. We want to find the probability that 5 of the next 7 students to arrive will be undergraduates, given that the professor is currently talking with an undergraduate.

We can use the binomial distribution to calculate this probability. The binomial distribution describes the number of successes (in this case, undergraduate student arrivals) in a fixed number of independent Bernoulli trials.

The probability of exactly k successes in n trials, given the success probability p, is given by the formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

So, to find the probability that 5 of the next 7 students to arrive will be undergraduates, we need to know the success probability p for undergraduate student arrivals.

(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?

To calculate the expected length of time, we need to use the concept of conditional probability. We want to find the expected length of time until the next student arrives, given that the next student will be an undergraduate.

We can use the concept of inter-arrival times in a Poisson process. The inter-arrival times in a Poisson process follow the exponential distribution. The expected inter-arrival time in an exponential distribution with rate λ is given by the formula:

E(T) = 1 / λ

Where E(T) is the expected inter-arrival time, and λ is the arrival rate.

(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 department head arrived) include both an undergraduate and a graduate student?

To calculate the expected time, we need to use the concept of waiting times in a Poisson process. We want to find the expected time until the set of students observed by the department head includes both an undergraduate and a graduate student.

We can use the fact that the waiting time for the first occurrence of two events in two independent Poisson processes is equal to the minimum of two exponential distributions.

Let's define T1 as the waiting time for the first undergraduate student to arrive and T2 as the waiting time for the first graduate student to arrive. We need to find the expected value of min(T1, T2), which can be calculated using the formula:

E(min(T1, T2)) = 1 / (λ1 + λ2)

Where E(min(T1, T2)) is the expected time, λ1 is the arrival rate for undergraduate students, and λ2 is the arrival rate for graduate students.

By using these concepts and formulas, we can find the answers to each question.