Problem 4. (30 pts) The differential equation of failure rate (x) and the lifetime x (mileage for the car example) is given by A(x) = 1 dR(x) R(x) dx where R(x) is the reliability function versus mileage 0.01 for the timing belt of a specific car. x is x. Assume the failure rate is given by A(x) the car mileage in thousand miles. (a) Is (x) a DFR, CFR or IFR and why? 1.01 100 100/ (b) Show the procedure to develop the reliability function which is R(x)-R(0)e (c) Can you assume R(0)-1.0 and why? What type of distribution the failure time (mileage) has? The scale parameter -? and shape parameters -? Characteristic life=? (d) R(100)-? (c) MTTF-? (you can leave the gamma function as I'() in your answer) (f) If you want the Reliability, or probability that the timing belt is not broken on the road, to be at least 0.70 or 70%, by what mileage should the timing belt be changed?