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 λ = .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 fails-that 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 Aiꞌs, 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 λ. What type of distribution does X have?

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