1. Consider a hospital emergent room that opens 24 hours a day. Suppose that the arrivals of patients are well modeled by a Poisson Process with rate per hour.
a) A nurse starts her shift from 8am, what is the expected time to see her first patient in the day? If she starts a bit late, say 8:10am, answer the same question. (Please specify your assumptions.)
b) If she already seen 12 patients by 10am, what's the conditional probability of seeing 17 patients by 11 am?
c) If 80% of patients coming to the emergency room are treated and released, and the rest are admitted to the hospital for further treatment. What can we say about the admission process of the hospital?
d) Suppose there are two nurses in the emergent room, and the time that each nurse uses to treat a patient follows Exponential distribution with mean equal to 1/4 hours (Nurse Alice) and 1/5hours (Nurse. Bob) respectively. When Patient Cindy enters the emergent room at 10am., she finds that Alice is treating Patient Dough, and Bob is free. Cindy prefers to be treated by the woman nurse, Alice, but she does not know when Alice starts the treatment. What is the expected waiting time for Cindy?
e) Now suppose Cindy is treated by Bob. What is the probability that Cindy will leave the emergent room earlier than Dough?