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Which of the following is an advantage of multivariate correlational research over bivariate correlational research? O Bivariate correlational studies help address internal validity, whereas multivariate correlational studies do not. Multivariate correlational studies establish covariance, whereas bivariate correlational studies do not. O Some multivariate correlational studies help to address temporal precedence, whereas bivariate correlational studies do not. Multivariate correlational studies help to address external validity, whereas bivariate correlational studies do not.Problem 3. Consider the Markov chain shown in Figure 2. Figure 2: Problem 3 Markov chain 1. Let the initial distribution be Pr(A) : Pr(B) = 0.5. What is the probability distribution after one step? 2. What is the stationary distribution of the Markov chain? 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 1 0.2 0.8 0 0 regular? (Yes or No) (2) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0.1 0.4 0.5 0 1 regular? (Yes or No) (3) Is the Markov chain whose transition matrix whose transition matrix is 0 0 0.8 0 0.2 0.8 0.2 0 regular? (Yes or No) (4) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0.8 0 0.2 0 1 0 regular? (Yes or No)Consider a standard chessboard with an 8 x 8 grid of possible locations. We define a Markov chain by randomly moving a single chess piece on this board. The initial location Xo is sampled uniformly among the 82 = 64 squares. At time t, the piece then chooses Xt+1 by sampling uniformly from the set of legal moves given its current location Xt. For a description of legal chess moves, see: http://en. wikipedia. org/wiki/Rules_of_chess#Basic_moves. a) Suppose the chess piece is a king, which can move to any of the 8 adjacent squares. Is the Markov chain irreducible? Is the Markov chain aperiodic? b) Suppose the chess piece is a bishop. Is the Markov chain irreducible? Is the Markov chain aperiodic? c) Suppose the chess piece is a knight. Is the Markov chain irreducible? Is the Markov chain aperiodic?1. Write the dual for each of the following primal problems: Maximize u = -5x1 + 2x2 Minimize z = 6x1 + 3x2 Subject to: Subject to: -xitx2 s -2 6x1 - 3x2 + X3 2 2 2x1 + 3x2 $5 3x1 + 4x2 + X3 25 X1, X2 2 0 X1,X2,X3 20 Maximize u = -x1 + 2x2 - 3x3 Subject to: X1 - X2 5 -2 2x1 + X2 - X3 2 15 Xz + X3 = 10 X1, X2 2 0 X3 unrestricted 2. Find the optimal value of the objective function for the following problem by inspecting only its dual. Do not solve the dual by the simplex method. Explain how you found the answer. Minimize z = 10x1 + 4x2 + 5X3 Subject to: 5x1 - 7x2 + 3x3 2 50 X1,X2, X3 20 3. Consider the following set of inequalities. Note that X1 = X2 = 0 is not a feasible solution. A feasible solution can be found by using the trivial objective function (maximize z = x1 + X2) with the constraints and then solving the dual, from which a solution for the set of inequalities can be found. Find a feasible point with this method (show your work). 2x1 + 3x2 $ 12 -3x1 + 2x2 5-4 3x1 - 5x2 $ 2 x1 unrestricted; x2 2 0 . Consider the following LP. Use the dual problem to show that the basic solution with (x1, *2) = (0,0) is not optimal. Explain why. Maximize z = 2x1 + 4x2 + 4x3 - 3x4 Subject to: X1 + X2 + X3 = 4 X1 + 4x2 + X4 = 8 X1, X2, X3, X4 2 0
