Answer Happy

Problem 2. A community of N individuals is in the middle of a zombie outbreak. Fortunately, the community has managed to discover a serum that can cure zombies and prevents them from further infection. Suppose at the beginning of each day, every individual is in one of three possible conditions: non-infected, zombie, and cured. If during day t, a non-infected person becomes a zombie, he or she will remain a zombie the next day (t + 1), but will get cured from the following day (t + 2) onwards. Let Xt and Y4 denote the number of zombies and the number of non-infected persons on day t. During each day, the probability that a given non- infected person comes in contact with a given zombie is 0 < p < 1, independently for every zombie and every other non-infected person. If this does happen, the non-infected person turns into a zombie for day (t + 1), and gets cured on day (t + 2). (a) If Xt = i, what is the probability that a given non-infected person will come in contact with a zombie during day t? (b) Is the pair (Xt, Yt), t = 0,1,2, ..., a Markov chain? If so, give an expression for its state space and transition probabilities. (c) Suppose Xo = 1, Y0 = N – 1, find the distribution of X2. Leave your answer as notation. =