2. In a clinic with two physicians, Dr. X and Dr. Y, each doctor can be scheduled to see at most 3 patients per hour. The joint probability mass function is given by the following table. a.) State two marginal probability mass functions pX and pY. b.) If we were to assume that Dr. X will see 2 patients in a given hour, what is the probability that Dr. Y will see exactly 1 patient? c.) If we were to assume that Dr. Y will see 0 patients in a given hour, what is the probability that Dr. X will see exactly 1 patient? d.) Is the number of patients seen in an hour by Dr. Y independent of the number of patients seen by Dr. X? Prove your answer. e.) What are the average and standard deviation in the number of patients seen per hour by Dr. X? f.) What are the average and standard deviation in the number of patients seen per hour by Dr. Y? g.) What is the probability that in any given hour Dr. X will see more patients than Dr. Y? Answers: 1 2 (Over ) 3. (Given three independent, normally distributed random variables, X1, X2, and X3, with means and standard deviations as follows, X1 = 4.00, X1 = 0.40, X2 = 2.00, X2 = 0.20, X3 = 5.00, X3 = 0.50, you need to study a random variable Y defined as Y = 2X1 + 1.5X2 - X3. a.) What is the mean of random variable Y? b.) What is the standard deviation of random variable Y? c.) Is Y normally distributed? Why or why not? d.) What is the probability that any given outcome for Y will be a value between 4.02 and 7.98? Answers: 3