S-72_2410_solution_2

# S-72_2410_solution_2 - S-72.2410 Information Theory Haanp...

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Unformatted text preview: S-72.2410 Information Theory Haanp aa & Linja-aho Homework 2, solutions 2009 Homework 2, solutions Deadline: November 16th, 16:00 The box for returning exercises is in the E-wing, 2nd floor corridor. 1. Let X and Y be two independent integer-valued random variables. Let X be uniformly distributed over { 1 , 2 ,..., 8 } , and let Pr { Y = k } = 2- k ,k = 1 , 2 , 3 ,... (a) Find H ( X ) (b) Find H ( Y ) (c) Find H ( X + Y,X- Y ). Solution: (a) For uniform distribution H ( X ) = 8(- 1 8 log 1 8 ) = log 8 = 3 (b) H ( Y ) =- k =1 1 2 k log 1 2 k = 1 2 k k log 2 = k =0 k ( 1 2 ) k = 1 2 (1- 1 2 ) 2 = 2 (c) Because ( X,Y ) ( X + Y,X- Y ) is a one-to-one -transformation, H ( X + Y,X- Y ) = H ( X,Y ). Because X and Y are independent, H ( X,Y ) = H ( X ) + H ( Y ) = 3 + 2 = 5. 2. A deck of n cards in order 1 , 2 ,...,n is provided. One card is removed at random then replaced at random. What is the entropy of the resulting deck? Solution: There are n 2 possible ways to shuffle the deck as described, and all have the same probability. But the solution is not that simple, because some shuffling actions result in same deck. There are basically three ways to shuffle the deck: The card is removed and put back in the same place. In this case,The card is removed and put back in the same place....
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## This note was uploaded on 10/27/2010 for the course ECE 221 taught by Professor Sd during the Spring '10 term at Huston-Tillotson.

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S-72_2410_solution_2 - S-72.2410 Information Theory Haanp...

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