6 Duration of life as a random variable
In the case of a live birth, i.e. if F
∗
(0+) = 1, and we agreed to consider
only such cases, we see that the limit of E[ T |T ≥ x] as x ↓ 0 equals
(1.4).
It is worth noting that as a function of x the expected duration of
life E[ T |T ≥ x] is increasing. If, for example, expected duration of
life at birth is 75 years, then for a person who lives to the age of 30
years, it is greater, and for a person who reached the age of 52 years,
it is still greater. How much greater? That depends on the form of the
distribution function F.
Exercise. Show that (1.11) defines an increasing function of x.
Use the fact that F
∗
(x) is non-increasing and do not worry about
differentiation under integral sign. 4
Exercise. Derive the ...