3. Suppose that the lifetime T; has hazard function h(t) and that C is a random censoring time associated with Ti. Define X(t) = lim At0 P(t T; t + t|Ti t, Ci t) (a) Show that if T; is independent of Ci, hi(t) = i(t). (b) Suppose that there exists an unobserved covariate Zi which affects both Ti and Ci, as follows: P(T; t|Z;) = exp(Z0t), P(C; t|Zi) = exp(-Zipt), and T, C are independent, given Zi. Assume that Z; has a gamma distribution with density ez (z > 0). Show that the joint survivor function for Ti, C is function g(z) = F(6) 2 88 1 8) = (1 + 1/06 (1+ + 100)" P(T; t, C; s) = ( :