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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?3. Suppose X has the pdf 0 otherwise (a) Find the cdf F. (b) Compute P(X (X - ]) (X - [) (X -1) > 0). (c) Compute P (X > (X > 1). (d) Compute P((x - ])'s :) (e) Compute E (X) and Var (X). (f) Use the Chebyshev Inequality to provide a lower bound on the probability you found in (d). Compare the bound to the value you computed. (g) Find E (X ) for k = 1, 2,3, 4, .... (h) Find P(max (X, 1 - X) > 2 min (X, 1 - X])