Question

(1 pt) Consider the transition probability matrix, P, below. 2 1- P 2 2 When solving for steady-state probabilities, it is necessary to solve ', subject to Ži ni- What are the equations describing this system? Note, use the variables p1 and p2 instead of Ti and T2. In steady state, T2 Note: Your answers must be in exact form

Answer 1

p = [2 p p^2 1 - 2 p 1 - p^2] p^2 = [2 p p^2 1 - 2 p 1 - p^2] [2 p p^2 1 - 2 p 1 - p^2] = [4 p^2 + p^2 - 2 p^3 2p^3 + p^2 - p^4 2 9 - 4 p^2 + 1 - p^2 - 2 p + 2 p^3 p^2 - 2 p^3 + 1 + p^4 - 2 p^2] p^2 = [5 p^2 - 2 p^3 p^2 + 2 p^3 - p^4 1 - 3 p^2 + 2 p^3 1 - p^2 - 2 p^3 + p^4] p^2 = [5 p^2 - 2 p^3 p^2 + 2 p^3 - p^4 1 - 5 p^2 + 2 p^3 1 - p^2 - 2 p^3 + p^4] Let = [p_1, p_2] [p_1, p_2] [2 p p^2 1 - 2 p 1 - p^2] = [p_1, p_2] Rightarrow [2 p p_1 + p_2 p^2 p(1 - 2 p) + p_2 (1 - p^2)] = [p_1, p_2] Rightarrow {2 p p_1 + p^2 p_2 = p_1, p_1 (1 - 2 p) + p_2 (1 - p^2) = p_2 and p_1 + p_2 = 1 p_1 = 1 - p_2} Therefore 2 p(1 - p_2) + p^2 p_2 = p_1 = 1 - p_2 2 p - 2 p p_2 + p^2 p_2 + p_2 = 1 p_2 (1 - 2 p p^2) = 1 - 2 p p_2 = 1 - 2 p/1 - 2 p + p^2