A system consists of five identical components connected in series as shown: As...

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 Ai 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 { } is equivalent to what event involving ?

b. Using the independence of the A_i’s , 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?