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10. In summarizing the relationship between two variables arranged in a bivariate table, explain how one goes about comparing the percentages in bivariate table if the independent variable has more than two categories. Correct Answer:2. A Markov chain with state space {1, 2, 3} has transition probability matrix 00 0.3 0.1 a: 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (c) Consider a general three-state Markov chain with transition matrix 3011 3012 1013 P = P21 P22 P23 1031 P32 P33 Give an example of a specic set of probabilities jag-'3; for which the Markov chain is not irreducible (there is no single right answer to this1 of course l]. For each of the following transition matrices, determine whether the Markov chain with that transition matrix is regular: (1) Is the Markov chain whose transition matrix whose transition matrix is 0 0.5 0.5 0.5 0 0.5 0 0 regular? (Yes or No) (2) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0 0.7 0 0 regular? (Yes or No) (3) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.6 0 0.4 1 0 0 regular? (Yes or No) (4) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0 0.6 0 0.4 regular? (Yes or No) (5) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0.2 0.5 0 1 02. 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 chain