Problem 4 Consider a random walk over three states {1. 2. 3}. At each state, with proba- I:Iiilit:,,r 1,32, the particle stajllrsl and 1with probability 1.34, the particle moves to one of the other two states. Let X; be the state of the particle at time t. Assume X0 = 1. {1} Write down the transition matrix T. {2} Let ptj = P{X1 = 1'), and let p. = {pg[1],p(2).p;[3}) be the row vector. Explain PHI = HT- (3) Based on [2}. calculate p\" for t = 1,2,3. (4) Calculate PLAT: = Ella = 1]. {5) Suppose 1 million particles do the random walk independently. Interpret your results in [3] and {4) in terms of proportions