The purchaser of a power-generating unit requires c consecutive successful start-ups before the unit will be accepted. Assume that the outcomes of individual start-ups are independent of one another. Let p denote the probability that any particular start-up is successful. The random variable of interest is X = the number of start-ups that must be made prior to acceptance. Give the pmf of X for the case c= 2. If p =.9, what is P(X ? 8)? [Hint: for x ? 5, express p(x) “recursively” in terms of the pmf evaluated at the smaller values x - 3, x – 4,……,2.] (This problem was suggested by the article “Evaluation of a Start-Up Demonstration Test,” J. Quality Technology, 1983: 103–106.)

Problem 105E Answer: Step1: We have the purchaser of a power-generating unit requires c consecutive successful start-ups before the unit will be accepted. Assume that the outcomes of individual start-ups are independent of one another. Let p denote the probability that any particular start-up is successful. The random variable of interest is X = the number of start-ups that must be made prior to acceptance. We need to give the pmf of X for the case c= 2. If p =0.9, what is P(X 8) [Hint: for x 5, express p(x) “recursively” in terms of the pmf evaluated at the smaller values x - 3, x – 4,……,2.] Step2: The probability of success of two start-ups is given by P(X = 2) = P(any particular start-up)×P(any particular start-up) = p×p 2 = p Similarly, The probability of success of three start-ups is given by P(X = 3) = P{(fail on 3rd start-up)×(success on...