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?
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(2) What is the expected length of time in hours that the 10th arriving student (undergraduate or graduate) will stay with the professor?
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(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?
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(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?
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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?
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(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?
To solve the given questions, we'll need to use the properties of the Poisson process and probability calculations. Let's go through each question step by step:
(1) To find the probability that exactly three undergraduates arrive between 10:00 pm and 10:30 pm, we can use the Poisson probability formula:
P(X=k) = (λ^k * e^(-λ)) / k!
Here, X represents the number of undergraduate students arriving in the given time interval. We're given that the rate of arrival for undergraduates is λu=3 per hour, and the time interval is 30 minutes (0.5 hours). So, we need to calculate P(X=3) using the formula.
(2) To find the expected length of time in hours that the 10th arriving student (undergraduate or graduate) will stay with the professor, we need to consider both undergraduates and graduates arriving at different rates. We'll use the concept of exponential distribution and cumulative distribution function (CDF) to calculate the expected time.
(3) Given that the professor is currently talking to an undergraduate student, we need to find the expected number of subsequent student arrivals up to and including the next graduate student arrival. This can be calculated using the conditional probability and the properties of the Poisson process.
(4) Similar to question (3), we need to find the probability that 5 out of the next 7 students to arrive will be undergraduates, given that the professor is currently talking to an undergraduate student. This involves conditional probability calculation using the Poisson process properties.
(5) 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'll use conditional probability and the properties of exponential distribution.
(6) Lastly, we need to find the expected time that the department head will have to wait until the set of students observed meeting with the professor includes both an undergraduate and a graduate student. This can be calculated using the properties of the Poisson process and conditional probability.
To solve these questions accurately, precise calculation and substitutions of the given rates and time intervals will be required.