4. (10 points) A homogeneous discrete-time Markov model has three states representing the status of the members of a population. 2 State 1 = healthy, no benefits State 2 = disabled, receiving Home Health Care benefits State 3 = disabled, receiving Nursing Home benefits (i) The annual transition probability matrix is given by 0.8 0.15 0.05 0.05 0.9 0.05 0 0 1 (ii) Transitions occur at the end of each year. (iii) At the start of year 1, there are 50 members, all in state 1, healthy. Calculate the variance of the number of those 50 members who have received Home Healthy Care benefits during the first three years