Question 4 (20 points) A large hospital faces uncertainty in the number of daily emergency department (ED) cases that it receives. Using its historical data, the hospital models the number of daily ED cases as a random variable X, With the following distribution: :1: P(X : I) 0 0.002 1 0.015 2 0.045 3 0.039 4 0.134 5 0.161 6 0.161 7 0.138 3 0.103 9 0.069 10 0.041 11 0.023 12 0.011 13 0.005 14 0.002 15 0.001 The number of ED cases each day is independent of the number of ED cases on any other day. (a) What is the probability that in a day, there are more than 10 ED cases? (1 points) (b) What is the probability that in a day, there are at least 5 and at most 8 cases? (1 points) (c) What is the expected value for the number of ED cases in a day? (2 points) (d) \"That is the standard deviation for the number of ED cases in a day? (2 points) (e) \"That is the probability that over a 60 day period, the hospital receives 300 cases or more in total? (Hint: What does receiving at least 300 cases in total over 60 days imply about the daily average number of cases?) (3 points) In order to ensure that the ED patients receive adequate care, the hospital staffs nurses. Each ED case requires a single nurse. Ideally, the hospital would be able to handle each ED case with a regular nurse, who is compensated at $2150 per day. The hospital sta's 8 regular nurses daily. These nurses must be paid, regardless of how many ED cases actually occur. However, if the number of ED cases exceeds 8, then the hospital needs to hire additional agency nurses to manage every case in excess of 8. These nurses are employed outside of the hospital, and are compensated at $5400 per day. Like regular nurses, each agency nurse can only manage one ED case. As an example, if there are 4 ED cases, then the total cost is 8 X $2160 = $17230 because the 8 regular nurses are staffed and must be paid, regardless of how many ED cases there are. As another example, if there are 10 ED cases, then the total cost is 8 x $2160 + (10 8) x $5400 : $28080 where there is an additional cost of two agency nurses to handle the two ED cases in excess of the 8. (f) What is the probability that on a given day, the hospital needs to hire at least one agency nurse? (3 points) (g) What is the expected value of the daily cost of the hospital's stafng plan? (4 points) (h) What is the probability that the hospital stas at least one agency nurse on 15 or more days out of the next 60 days? (4 points)