Thanks in advance for your help.

2. Consider a random walk X\" that Mario is taking on a nite state space: X,1 = {0, 1, 2, 3, 4, 5, 6}. The transition probabiiities are pjj+1 = 0.5,pj3-_1 = 0.5,j = 1,2, - -- ,5. Assume an ab- sorbing boundary condition on the left end, i.e., p0,.) = 1, and a reecting boundary condition on the right end, i.e., 30315 = 1. (a) (5 points] What is the 3step 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 no, why? 7. Consider \"Mario's jump\" in Question 2. Write a code to simulate this random walk. (a) (10 points) Generate 5 different paths. Each of them consists of 20 \"jumps\". Plot these 5 paths in one gure. (b) (10 points) Verify your answer to the following question (numerically): Is there any transient state? If yes, what is it? If no, why? (Hint: use the denition of \"transient\" and the computer simulation is a brutal force approach to investigate it.)