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Unformatted text preview: that the columns add to one, so the stationary distribution is uniform. To check the hypothesis of the convergence theorem, we note that after 3 turns we will have moved between 3 and 9 spaces so p3 (i, j ) > 0 for all i and j . Example 1.27. Mathematician’s Monopoly. The game Monopoly is played on a game board that has 40 spaces arranged around the outside of a square. The squares have names like Reading Railroad and Park Place but we will number the squares 0 (Go), 1 (Baltic Avenue), . . . 39 (Boardwalk). In Monopoly you roll two dice and move forward a number of spaces equal to the sum. For the moment, we will ignore things like Go to Jail, Chance, and other squares that make the game more interesting and formulate the dynamics as following. Let rk be the probability that the sum of two dice is k (r2 = 1/36, r3 = 2/36, . . . r7 = 6/36, . . ., r12 = 1/36) and let p(i, j ) = rk if j = i + k mod 40 where i + k mod 40 is the remainder when i + k is divided by 40. To explain suppose that we are sitting on Park Pl...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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