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

F(t; l, n) 5 P(X # t)

1 2345 As soon as one component fails, the entire system will fail.

Suppose each component has a lifetime that is exponentially distributed with and that components fail independently of one another. Define events {ith component lasts at least t hours}, , so that the are independent events. Let 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 , compute .

Then obtain 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?