For a health insurance plan, for every client, the company distinguishes three conditions at the beginning of each year: “healthy”, “sick”, and “deceased”. The transition matrix is

Assume that at the initial moment, 90% of the clients of the company are healthy.

(a) Find the expected lifetime of a client selected at random.

(b) Assuming that an insured was healthy at the beginning and not counting the initial state, find the expected length of the period when the insured remained healthy before switching to another state.

(c) Suppose that the mean annual health care cost for the two first states—“healthy” and “sick”—equals 1 and 4 (units of money), respectively. Find the expected present value of the total cost (i.e., the expected discounted total cost) in the long run with a discount of 0.95. (Advice: Assign zero payment to the absorbing state “deceased.”)