Question

5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive component in a system, which is exponentially distributed with mean 2 years. The system also has a cheaper backup component that can take over when the expensive component fails so that the system can provide continuous service while the more expensive system is being repaired. Let Y be the lifetime of the backup system, which is also exponentially distributed but with a shorter mean 0.5 years. Assume that the lifetimes of the expensive and backup components are independent. (a) Suppose that it turns out that the expensive system cannot be repaired. Find the pdf of the total lifetime of the system Z = X + Y (b) Let U -X - Y be the difference in the lifetimes of the two systems. Note that U can be both positive and negative, since it is possible (although unlikely) that the more expensive component may last for less time than the backup. Find the pdf of U

Answer 1

uite at bact at component (2-t) e x dt 2. e dk 2e

- 2 (x-u) e 2 d A x 극e eaa 一2-5