A system consists of five identical components connected in series as shown: 2 5 As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with A= .01 and that components fail inde- pendently of one another. Define events 4, = {ith compo- nent lasts at least hours), i = 1,..., 5, so that the As 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} is equivalent to what event involving AA?

b. Using the independence of the 4's, compute P(X = 1). Then obtain F(1) = P(X) and the pdf of X. What type of distribution does X have?

c. Suppose there are n components, each having exponen- tial lifetime with parameter A. What type of distribution does X' have?