15. Consider a system consisting of three components as pictured. The system will continue to function as long as the rst component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the lifetimes of components 1, 2, and 3, respectively. Suppose the Xi s are independent of one another and each Xi has an exponential distribution with parameter l.

5.1 Jointly Distributed Random Variables 241 1

3 2

a. Let Y denote the system lifetime. Obtain the cumulative distribution function of Y and differentiate to obtain the pdf. [Hint: F( y)  P(Y y);

express the event {Y y} in terms of unions and/or intersections of the three events {X1 y}, {X2 y}, and {X3 y}.]

b. Compute the expected system lifetime.