Consider a system consisting of three components as pictured. The system will continue to function as long as the f(x, y) 5 e xe2x(11y) x $ 0 and y $ 0 0 otherwise g

m k50 a

m k

bak bm2k 5 (a 1 b)m 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 Xi

’s are independent of one another and each Xi has an exponential distribution with parameter l.

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.