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, F¯

t

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

a. Show that

(a) F¯

t

(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 F¯

t

(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.