ps1_sol_10_11 - MS&E 221 CA Hongsong Yuan Problem 1(a E[Xn...

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MS&E 221 Problem Set 1 Solutions CA: Hongsong Yuan January 19, 2011 Problem 1 (a) E [ X n ] = 0 (b) Var[ X ] = Var[ E [ X | P ]] + E [Var[ X | P ]] Var[ X n ] = Var[ E [ X n | X n - 1 ]] + E [Var[ X n | X n - 1 ]]] = 0 + E [ βX 2 n - 1 ] = β E [ X 2 n - 1 ] = β (Var[ X n - 1 ] + ( E [ X n - 1 ]) 2 ) = β Var[ X n - 1 ] Var[ X 1 ] = βx 2 0 Var[ X n ] = β n x 2 0 Note: The sequence is not necessarily a Markov chain. The current mean and variance just depend on the previous step, but the distribution itself could depend on all previous steps. Problem 2 Let Ω be the probability space and A, B, C be 3 disjoint subsets that evenly divides Ω , that is, P ( A ) = P ( B ) = P ( C ) = 1 3 . Define X, Y, Z as follows: X = - 2 I ( A ) + I ( B ) + I ( C ) Y = 0
Problem 3 For each part, I’ll give both the reason for modeling the process as a Markov chain and the reason for not doing so. 1