Homework Solution 1

Homework Solution 1 - HMWK 1 1 3 black balls and 3 white balls are distributed among two urns(labeled 1,2 At each “move” a ball is randomly

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Unformatted text preview: HMWK 1 1. 3 black balls and 3 white balls are distributed among two urns (labeled 1,2). At each “move”, a ball is randomly selected from each urn and the two are swapped (interchanged). Let X n denote the number of Black balls in Urn 1 after the n th move. Argue that X n forms a Markov chain, and give the transition probability matrix ( P i,j ). Solution: The state space is S = { , 1 , 2 , 3 } . We assume that each urn contains 3 balls each. Given X n = i ∈ S , we know that the first urn contains i black ( B ) balls and 3- i white ( W ) balls, whereas the second urn contains 3- i B balls and i W balls. We thus know the complete distribution of the balls among the urns if we know the state of X n . Randomly selecting one ball from each urn and swapping them, indeed yields a MC: If X n = i ∈ { , 1 , 2 , 3 } , then, independent of the past, the ball chosen from the first urn will be B wp= i/ 3, and W wp= (3- i ) / 3, whereas (independently) the ball chosen from the second urn will be B wp= (3- i ) / 3, and W wp= i/ 3. This leads to the following transition probabilities: P 01 = 1 , P 10 = 1 9 , P 21 = 4 9 , P 32 = 1 , P 11 = 4 9 , P 22 = 4 9 , P 12 = 4 9 , P 23 = 1 9 . P = 1 1 9 4 9 4 9 4 9 4 9 1 9 1 . For example P 10 = 1 / 3 × 1 / 3 = 1 / 9 because this results from the events : Chose B from the first urn and chose W from the second urn. P 22 = 4 9 because either a W was swapped with a W (wp= 2 / 9), or a B was swapped with a B (wp= 2 / 9), yielding a total (sum) of 4 / 9. 2. Consider modeling the weather where we now assume that the weather today depends (at most) on the previous three days weather. Letting W n denote weather on the n th day (0 = no rain, 1 = rain), let X n = ( W n- 2 ,W n- 1 ,W n ). Argue that { X n } forms a MC. There are 8 states, and we can relabel them 0—7 (as was pointed out in lecture)....
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This note was uploaded on 10/16/2010 for the course IEOR 4701 taught by Professor Karlsigma during the Spring '10 term at Columbia.

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Homework Solution 1 - HMWK 1 1 3 black balls and 3 white balls are distributed among two urns(labeled 1,2 At each “move” a ball is randomly

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