Question in the picture below

(1) (4 points) Suppose that Tom carries Covid19 virus but asymptotic. He starts at node 2 on day 0. On each day, he moves randomly to one of the neighboring nodes. Let X\" be Tom's location (node) on day n. It is known that X = {Xn : n = 0, 1, 2,. . .} is a DTMC on S = {1, 2, 3,4,5, 6, 8,9}. Is X irreducible? Is X aperiodic? What is the period of X? (2) (4 points) Suppose that Lisa is virus free. She starts at node 9 on day 0. On each day, she moves randomly, independently of Tom, to one of her neighboring nodes. Let Yn be Lisa's location (node) on day R. If Lisa runs into Tom at a (common) node, Lisa is infected. What is the the probability that Lisa is rst infected on day I: for k = 1, 2. (3) (4 points) Suppose that there are testing sites at node 1, node 6, node 8 and node 9. Assume that Tom is currently at node 5. What is the probability that he gets tested at site 1 before at sites 6, 8 or 9? (4) (4 points) Prove that {(Xn, Y\") : n = 0, 1, 2, . . .} is also a DTMC. Spell out its state space. (5) (4 points) Is {(Xn, Y\") : n = 0, 1, 2, . . .} irreducible? Prove your claim. (6) (4 points) Tom is initially at 1 and Lisa at 7. What is the probability that the Lisa will eventually be infected? Write a number is not sufcient. You must support your claim