6.32 Let X1, X2, . .., Xn be a random sample from a uniform distribution on the interval [0, 0], which has mean u = 0/2 and variance o2 = 02/12. Consider two estimators of 0: 01 = X max and 02 = 2X. (a) Use the result of Exercise 5.44(b) to find E (01), Var(01), and MSE(01). () Find E(02), Var(02), and MSE(02). (c) Compare MSE(01 ) and MSE (02).5.44 Consider n identical components whose lifetimes, Ti, T2, ..., In are i.i.d. uniform r.v.'s over the interval [0, 0], where 0 is an upper limit on the lifetime. (a) Suppose that the components are connected in series. Show that the lifetime of the system is T(1) = Tmin with p.d.f. f (1 ) = n Use this p.d.f. to show that the mean time to failure (MTTF) of the system is E(Tmin) = n + 1 (b) Suppose that the components are connected in parallel. Show that the lifetime of the system is T(n) = Tmax with p.d.f. n - f (n) = n Use this p.d.f. to show that the MTTF of the system is E(Tmax) = n n + 1 (c) Compare the MTTF's of the series and parallel systems as n increases