A dedicated professor has been holding infinitely long office hours. Undergraduate students arrive according to a Poisson process at a rate of λu=3 per hour, while graduate students arrive according to a second, independent Poisson process at a rate of λg=5 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?
(2) What is the expected length of time in hours that the 10th arriving student (undergraduate or graduate) will stay with the professor?
(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?
(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?
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?
(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?
anyone please provide answers to above problem.
1. 0.125
2. 1/8
3. 1.5
4. 0.0608
anyone has answer for 5?
3. 1.5 is incorrect :(
Ans for 3, 5 and 6? Anyone?
1. 0.125
2. 1/8
3. ???
4. 0.0608
5. 1/8
6. ???
Any help?!
6. 34/120
3. 1.6
To solve these problems, we will be using the concepts of Poisson processes and conditional probability.
1) The probability that exactly three undergraduates arrive between 10:00 pm and 10:30 pm can be calculated using the Poisson distribution formula.
The formula for the Poisson distribution is P(X = k) = (e^(-λ) * λ^k) / k!, where λ is the average rate of arrivals.
In this case, λu = 3 since undergraduates arrive at a rate of 3 per hour. To calculate the probability of exactly three arrivals, we substitute k = 3:
P(X = 3) = (e^(-3) * 3^3) / 3! = (e^(-3) * 27) / 6
2) To find the expected length of time that the 10th arriving student will stay with the professor, we need to consider the sum of the waiting times for the previous 9 students.
Since both undergraduates and graduates arrive independently as separate Poisson processes, we can calculate the expected length of time for each type of student separately and then add them together.
The expected waiting time for the 10th undergraduate student can be calculated using the formula E(X) = 1 / λ, where λu = 3.
E(undergraduate) = 1 / 3
Similarly, the expected waiting time for the 10th graduate student can be calculated using the formula E(X) = 1 / λ, where λg = 5.
E(graduate) = 1 / 5
The expected length of time for the 10th arriving student is the sum of these two expectations:
E(10th student) = E(undergraduate) + E(graduate)
3) Given that the professor is currently talking with an undergraduate, we can use the concept of conditional probability to find the expected number of subsequent student arrivals up to and including the next graduate student arrival.
Let's denote this expected value as E(X), where X represents the number of subsequent student arrivals. We need to consider that the professor is occupied until the next graduate student arrives, hence we subtract one from X.
To calculate E(X), we can multiply the average time between graduate student arrivals, which is 1 / λg, by the rate of undergraduate student arrivals, which is λu.
E(X) = (1 / λg) * λu
4) Similarly, given that the professor is currently talking with an undergraduate, we can use conditional probability to find the probability that 5 of the next 7 students to arrive will be undergraduates.
This can be calculated using the binomial distribution formula. The formula for the binomial distribution is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the total number of trials, k is the number of successful outcomes, and p is the probability of success.
In this case, n = 7, k = 5, and p = λu / (λu + λg).
P(X = 5) = (7 choose 5) * ((λu / (λu + λg))^5) * ((λg / (λu + λg))^(7-5))
5) Beginning at midnight, to find the expected length of time until the next student arrives, conditioned on the event that the next student will be an undergraduate, we can use conditional probability.
Let's denote this expected value as E(X), where X represents the waiting time until the next student arrives. We need to consider that the next student will be an undergraduate, hence the rate of arrivals would be λu.
E(X) = 1 / λu = 1 / 3
6) To find the expected time that the department head will have to wait until the set of observed students includes both an undergraduate and a graduate student, we can use the concept of conditional probability and the waiting time for each type of student.
Let's denote this expected value as E(X), where X represents the waiting time until there is at least one undergraduate and one graduate student in the observed set. We can calculate E(X) by summing the expected waiting times for observing an undergraduate student and a graduate student separately.
E(undergraduate) = 1 / λu = 1 / 3
E(graduate) = 1 / λg = 1 / 5
E(X) = E(undergraduate) + E(graduate)