77. 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 l  .01 and that components fail independently of one another.

Define events Ai  {ith component lasts at least t hours}, i1, . . . , 5, so that the Ai’s are independent events. Let Xthe 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 five Ai’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 l. What type of distribution does X have?