hw2soln - EE 670 Homework#2 Solution Prof Uf Tureli Stevens...

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EE 670 : Homework #2 Solution Prof. Uf Tureli, Stevens Institute of Technology email/tel/fax: [email protected], 201.216.5603/8246 Note: Cover and Thomas, Q. 3.4,4.2,4.7,4.10 1. Question: Products. Let X = 1 , with probability 1 2 2 , with probability 1 4 3 , with probability 1 24 (1) Let X 1 , X 2 , ··· be drawn i.i.d. according to this distribution. Find the limitiing behavior of the product ( X 1 X 2 ··· X n ) 1 n . Solution : Let P n = ( X 1 X 2 ··· X n ) 1 n . Then: log P n = 1 n X i = 1 n X i X i , with probability 1 by strong law of large numebr. Thus, P n 2 E log X with prob. 1 and E log X = 1 2 log 1 + 1 4 log 2 + 1 4 log 3 = 1 4 log 6. P n 2 1 2 log 6 . 2. Question: Time’s arrow. Let { X i } i = -∞ be a stationary stochastic process. Prove: H ( X 0 | X - 1 , X - 2 , ··· , X - n ) = H ( X 0 | X 1 , X 2 , ··· , X n ) Solution : By chain rule for entropy: H ( X 0 | X - 1 , X - 2 , ··· , X - n ) = H ( X 0 , X - 1 , X - 2 , ··· , X
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hw2soln - EE 670 Homework#2 Solution Prof Uf Tureli Stevens...

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