77. A system consists of five identical components connected in series as shown: 2 3 As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with 2 = .01 and that components fail independently of one another. Define events A; = {ith component lasts at least 1 hours}, i = 1, . , 5, so that the A,'s are independent events. Let X = the time at which the system fails- that is, the shortest (minimum) lifetime among the five components. a. The event {X > t} is equivalent to what event involving A1, . .. , Ag? b. Using the independence of the five A¡'s, com- pute P(X > 1). Then obtain F(t) = P(X < 1) and the pdf of X. What type of distribution does X have? c. Suppose there are n components, each having exponential lifetime with parameter â. What type of distribution does X have?

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