Question 9.EX.22: Let X denote the lifetime of an item. Suppose the item has r......
Let X denote the lifetime of an item. Suppose the item has reached the age
of t. Let X_{t} denote its remaining life and define
\bar{F_{t}} (a) = P\left\{X_{t} >a\right\}
In words, \bar{F_{t}} (a) is the probability that a t-year old item survives an additional time a. Show that
(a) \bar{F_{t}} (a) = \bar{F}(t +a)/ \bar{F}(t) where F is the distribution function of X.
(b) Another definition of IFR is to say that F is IFR if \bar{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.
(a)\bar{F}_t(a) =P\{X>t+a \mid X>t\}
=\frac{P[X>t+a\}}{P\{X>t\}}=\frac{\bar{F}(t+a)}{\bar{F}(t)}
(b) Suppose λ(t) is increasing. Recall that
\bar{F} (t)= e^{-\int_{0}^{t}{} λ(s)ds}Hence,
\frac{\bar{F}(t+a)}{\bar{F}(t)}=\exp \left\{-\int_t^{t+a} \lambda(s) d s\right\}
which decreases in t since \lambda(t) is increasing. To go the other way, suppose \bar{F}(t+a) / \bar{F}(t) decreases in t. Now when a is small
\frac{\bar{F}(t+a)}{\bar{F}(t)} \approx e^{-a \lambda(t)}
Hence, e^{-a \lambda(t)} must decrease in t and thus \lambda(t) increases.