The lifetime of a driver circuit at the front end of device is modeled as Y and the lifetime of the device is modeled as X. It is obvious that the device will cease to operate when the driver circuit fails while the converse is not true. The device may fail even when the driver circuit is working. The joint density of these two life times is: f(x, y)= K exp(-y/10), 0 < x < y< [infinity]

Obtain the value of K so that f(x, y) is a valid density. Obtain the mean life times of the driver circuit, device. If the device fails, what is the expected lifetime of the driver circuit?