*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 Ft

(a) = P{Xt > a]

In words, Ft

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

a. Show that

(a) Ft

(a) = Fit + 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 ty for all

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