A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X= the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of ?X ?is a.? ?Find the value of ?k ?and draw the corresponding density curve. [?Hint?: Total area under the graph of ?f?(? is 1.] b.? ?What is the probability that the lecture ends within 1 min of the end of the hour? c.? ?What is the probability that the lecture continues beyond the hour for between 60 and 90 sec? d.? ?What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?

Problem 5E Answer: Step1: We have A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X= the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X s We need to find, a. Find the value of k and draw the corresponding density curve. [Hint: Total area under the graph of f(x) is 1.] b. What is the probability that the lecture ends within 1 min by the end of the hour c. What is the probability that the lecture continues beyond the hour for between 60 and 90 Sec d. What is the probability that the lecture continues for at least 90 Sec beyond the end of the hour Step2: a). Consider, 2 F(x) = kx Integrate above equation with respect to x then we get by taking limits from 0 to 2 2 2 2 P(X 2) = f(x)x = (kx )dx 0 0 2 = k (x )dx 0 2+1 = k x | 231 0 = k x | 3 0 3 = k (0+ 3 ) 8 = k 3 3 k = 8 Therefore, the value of k is 0.3750. b). Consider, 2 F(x) = kx Integrate above equation with respect to x then we get by taking limits...