Question

A health researcher is studying obesity in a community. She first divides people into three weight groups: thin, normal, and overweight. By studying the eating behavior of the population as well as the characteristics of the community, she finds that a thin person has 50% chance to become normal and 20% chance to become overweight after a year. A normal weight person has 20% chance to become thin and 20% chance to become overweight after a year. And an overweight person has 10% chance to become thin and 50% chance to become normal after a vear a) Form the one-step transition probability matrix P b) For a normal weight person: what is the probability of being in normal weight after 2 years? what is the probability of becoming thin after 2 years? what is the probability of being overweight after 2 years. c) Suppose that at present 20% of people are thin, 55% are normal, and 25% are overweight. What percentage of people are overweight 2 years from now? What percentage of people are NOT overweight five years from now?

Answer 1

a) one step transition probability matrix p = [0.3 0.2 0.1 0.5 0.6 0.5 0.2 0.2 0.4] b) find p^2 to observe behaviour after 2 years : p^2 = [ 0.21 0.2 0.17 0.55 0.56 0.55 0.24 0.24 0.28] p_r (N rightarrow N) (element (2 , 2) of p^2 ) = 0.56 p_r (N rightarrow T) (element (2 , 1) of p^2) = 0.20 p_r (N rightarrow D) (element (2 , 3) of p^2) = 0.24 c) present state is represented by the matrix x = [0.2 0.55 0.25] after 2 yrs , the state would be y = xp^2 = [0.2 0.55 0.25 ] [0.21 0.20 0.17 0.55 0.56 0.550.24 0.24 0.28] = [ 0.1945 0.5555 0.25] % no of overweight people 2 years from now = Y(3) = 0.25 times 100 = 25 % after 5 years , state would be Y = XP^5 = [0.19444 0.5556 0.25 ] therefore % of people NOT overweight 5 years from now = (1 - 0.25) times 100 = 75 %