S-72_2410_solution_1

S-72_2410_solution_1 - S-72.2410 Information Theory Haanpaa...

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Unformatted text preview: S-72.2410 Information Theory Haanpaa & Linja-aho Homework 1, solutions 2009 Homework 1, solutions Deadline: November 9th, 16:00 The box for returning exercises is in the E-wing, 2nd floor corridor. 1. Let p ( x, y ) be given by X downslope Y 1 1 3 1 2 1 1 6 Find: (a) H ( X ) , H ( Y ) . (b) H ( X | Y ) , H ( Y | X ) . (c) H ( X, Y ) . (d) H ( Y )- H ( Y | X ) . (e) I ( X ; Y ) (f) Draw a Venn diagram for the quantities in parts (a) through (e). Solution: (a) H ( X ) =- x X p ( x ) log p ( x ) =- p (0) log( p (0))- p (1) log( p (1)) =- 5 6 log 5 6- 1 6 log 1 6 . 65 H ( Y ) =- y Y p ( y ) log p ( y ) =- p (0) log( p (0))- p (1) log( p (1)) 1 (b) H ( X | Y ) = y Y p ( y ) H ( X | Y = y ) = 1 2 H ( X | Y = 0) + 1 2 H ( X | Y = 1) = 1 2 (- 1 3 log 1 3- 2 3 log 2 3 ) + 1 2 . 459 H ( Y | X ) = x X p ( x ) H ( Y | X = x ) = 5 6 H ( Y | X = 0) + 1 6 H ( Y | X = 1) = 5 6 (- 2 5 log 2 5- 3 5 log 3 5 ) + 1 6 . 809 (c) H ( X, Y ) =- x X y Y p ( x, y ) log p ( x, y ) =- 1 3 log 1 3- 1 6 log 1 6- 1 2 log 1 2- log 0 1 . 459 (d) H ( Y )- H ( Y | X ) = 1- . 809 = 0 . 191 (e) I ( X ; Y ) = H ( X, Y )- H ( X | Y )- H ( Y | X ) = 0 . 191 (or H ( Y )- H ( Y | X ) or H ( X )- H ( X | Y )) (f) See figure below: Page 1 of 4 S-72.2410 Information Theory Haanpaa & Linja-aho Homework 1, solutions 2009 H ( Y | X ) H ( X | Y ) H ( X ) H ( Y ) I ( X ; Y ) H ( X, Y ) 2. A fair coin is flipped until the first head occurs. Let X denote the number of flips required....
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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_1 - S-72.2410 Information Theory Haanpaa...

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