The Kapan-Meier estimate of the survival function is based on discrete time. Accordingly, let the survival time T be a discrete random variable with non-zero probability on the points t1, t2, . . . . Also, t0 = 0, and P(T = t0) = 0.

1 . Let p; = the probability of surviving past time t; , given survival to time t; _ 1 . That is , Pi = P ( I' > t; IT' > t; - 1 ). Prove Pj = S ( t; - 1 )| S" ( t; )