Let the survival time Τ > 0 be an integer-valued random variable with finite mean r0 and let…

Let the survival time Τ > 0 be an integer-valued random variable with finite mean r0 and let

be the expected residual life at time i, i = 1,2, Show that the survivor function for integer t is

Thus, in the discrete case also, the residual mean lifetime specifies the distribution of T. (Note: The geometric argument in Exercise 1.4 can be used to show that

Exercise 1.4

Let t be a discrete failure time variable taking values on the points with survivor function F(t). Show that the area under the survivor function,  (Note: A simple geometric proof of this is obtained by partitioning a plot of the survivor function into rectangles with bases along the vertical axis.)