The arrival times of students at the health center at a university is known to have a

Question:

The arrival times of students at the health center at a university is known to have a Poisson with an average rate of 6 sick students arriving per hour.
a. What is the expected number of students who arrive in the next 30 minutes?
b. What is the expected time that will elapse between the 3rd and 4th arrivals? What is the standard deviation?
c. What is the expected time that will elapse between the 4th and 7th arrivals? What is the standard deviation?
d. What is the probability that there will be at least 15 minutes between each of the next 3 arrivals?
e. What is the probability there will be fewer than 3 visits to the health center during the next hour?
f. The health center is open from 9 AM to 5 PM. What is the probability that there will be exactly one hour (9-10, 10-11, etc.) with strictly fewer than 3 visits?
g. Yesterday, a temporary secretary was working the admissions desk. He kept records on how many students visited each hour, but not their exact times of arrival. If we know exactly one student visited the health center during the 2-3 PM hour, what is the probability the student arrived within 5 minutes of 1:30?
h. The fraction of students who leave the health center each day with a prescription for little blue Amoxicillin pills is modeled by the following probability density function:

What is the expected fraction of students who will receive this prescription next week?

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