Let X denote the lifetime of an item. Suppose the item has reached the age of t .

Let Xt denote its remaining life and define Ft

(a) = P{Xt >a}
In words, ¯ Ft

(a) is the probability that a t-year-old item survives an additional time

a. Show that

(a) ¯ Ft

(a) = ¯ F(t +a)/ ¯ F(t) where F is the distribution function of X.

(b) Another definition of IFR is to say that F is IFR if ¯ Ft

(a) decreases in t , for all

a. Show that this definition is equivalent to the one given in the text when F has a density.