Random walk

2. Consider arandom walk X\" that Mario is taking on a nite state space: X" 2 {0,1, 2, 3, 4, 5, 6}. The transition probabilities are pjj+1 = 0.5,pj3-_1 = 0.5,j = 1,2, - -- ,5. Aguirre an ab sorbing boundary condition on the left end, i-e., pm = 1, and a reecting boundary condition on the right end, i.e., 1955 = 1. (a) (5 points) What is the 3-step transition matrix 13(3)? If Mario starts at location 2, what is the probability that he arrives at location 5 after 3 steps? (b) (5 points) Is there a limiting probability vector? If yes, what is it? If no, what is the period? (c) (5 points) Is there any invariant probability distribution? If yes, what is it? Is it unique? If no, why? (d) (5 points) Is there any recurrent state? If yes, what is it? If no, why? (e) (5 points) Is there any transient state? If yes, what is it? If 110, why