You went to a clinic to visit the doctor. There were 4 patients ahead of you in the waiting room. Their waiting times to meet the doctor were 12 minutes, 16 minutes, 12 minutes, and 8 minutes, respectively. Let X be the random variable for the waiting time of a patient. (a) Assuming X ~ Exponential (X) (i.e., an exponential distribution), use the maximum likelihood approach to estimate X from the waiting times of the 4 patients in front of you. (b) Based on your answer in part (a), what is the probability that your waiting time is less than 10 minutes? (c) Suppose the office has a sign that says the average wait time of a patient is about 10 minutes. Use this information to define the prior probability of A, i.e., A Exponential(). Based on this prior, calculate the value of λ using the maximum a posteriori (MAP) ap- proach. (d) Based on your answer in part (c), what is the probability that your waiting time is less than 10 minutes?

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