Answer: A system consists of five identical components

Chapter 4, Problem 4.77

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A system consists of five identical components connected in series as shown: As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with .01 and that components fail independently of one another. Define events Ai {ith component lasts at least t hours}, i 1, . . . , 5, so that the Ais are independent events. Let X the time at which the system failsthat is, the shortest (minimum) lifetime among the five components. a. The event {X t} is equivalent to what event involving A1, . . . , A5? b. Using the independence of the Ais, compute P(X t). Then obtain F(t) P(X t) and the pdf of X. What type of distribution does X have? c. Suppose there are n components, each having exponential lifetime with parameter . What type of distribution does X have?

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