Consider a system consisting of three components as pictured

Consider a system consisting of three components as pictured. The system will continue to function as long as the first 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 X_i\'s are independent of one another and each X_i has an exponential distribution with parameter .

(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},{X₃ y}.]

(b) Compute the expected system lifetime.

Solution