22. 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

¯F t

(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.