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Consider a Markov chain {Xn, n = 0, 1, . ..} on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 2 10 10 10 2 P = 3 10 2 4 10 10 1. Show that the Markov chain has a unique stationary mass. 2. Let h denote the stationary mass of the Markov chain. Find h(x) for all x E S. 3. Show that the Markov chain has the steady state mass. 4. Let h* denote the steady state mass of the Markov chain. Find h*(x) for all x E S.2. Markov chain transitions P = [P/j] = Al- AI- NI- Al- NI- AI- NI- AI- Al- Let X1 be distributed uniformly over the states {0, 1, 2}. Let (Xill be a Markov chain with transition matrix P; thus, P(Xn+1=j \\Xn= i) = Pu , i, j E {0, 1, 2}. (a) Is the information source stationary? ~ (b) Find the stationary distribution of the Markov chain (c) Find the entropy rate of the Markov chain1. Consider the Markov chain with state space {0, 1, 2} and transition matrix 1 2 P = HO 2 (a) Suppose Xo = 0. Find the probability that X2 = 2. (b) Find the stationary distribution of the Markov chain. (c) What proportion of time does the Markov Chain spend in state 2, in the long run? (d) Suppose X5 = 1. What is the expected additional number of steps (after time 5) until the first time the Markov chain will return to state 1?Determine if the following items represent an example of positive economics or normative economics. The richest 1% of Americans should pay more taxes than the rest of the 99%. O Positive Economics O Normative Economics A decrease in the supply of coconut will increase the price of German O Normative Economics chocolate cake, a good which requires coconut shavings as a key ingredient O Positive Economics The higher the minimum wage, the higher the price of goods and services Normative Economics is likely to bo. OO Positive Economics Social welfare spending in Sweden occupies too large a portion of the O Positive Economics national budget. O Normative Economics