1. An insurance policy provides a benefit of 1 at the moment of failure provided this occurs after 10 years. The hazard rate of the failure time is a constant 0.10 and the force of interest is a constant 0.05. If Z is the present value of the benefits, find the probability that: (a) 0 Z ≤ 0.5, (b) 0 ≤ Z ≤ 0.5 and (c) 0 ≤ Z ≤ 0.7.
2. An insurance policy provides a benefit of 1 at the moment of failure plus a pure endowment of 1 at time 10. This is purchased by level premiums payable continuously for 10 years. The premium is the net premium plus 10%. There is a constant force or mortality of 0.05 and a constant force of interest of 0.05. Find (a) P(-1/3 L ≤ 1/2) and (b) P(-1/4 ≤ L ≤ 1/2).