(5) l. A system consists of three components connected so that at least 2 components must operate for the system to operate properly, i.e. this is 2-out-of-3 system (see p. 137 # 99). Suppose that each component has a lifetime that is exponentially distributed with A = .001, and that components fail independently. Let X = the time at which the system fails, Xi = lifetime of -th component, i = 1, 2, 3. 3.) Calculate the cuff F (x) 2 P(X 5 m), the pdf of X and the expected value of X. b) Determine the probability that the system will not fail before 600 hours. Hint: Introduce rv Y(m) = # of components working after a: and relate F(:c) with pmf of Y(:e). Chapter 3, End of Chapter, SUPPLEMENTARY EXERCISES, Page 137 L p] 3 Exercise 99 A k-outofn system is one that will function if and only if at least k of the 71 individual components in the system function. If individual components function independently of one another, each with probability .9, what is the probability that a 3-out-of-5 system functions? We've got a stepbystep solution to this question! Dnu'unu. i'kn n+nnr- k. . nlinlzinn +kn In. I+l'r\ hnlnu