sample_quiz5_solution

sample_quiz5_solution - X and Y are indpendent) = 2...

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Math 187 Prof. Garsia typed by Alex Brik 05-11-2010 SAMPLE QUIZ 5 solution. 1. a. H [ X ] = P ( X = 5) log 2 1 P ( X =5) ± + P ( X = ± 3) log 2 1 P ( X = 3) ± + P ( X = 2) log 2 1 P ( X =2) ± = 1 4 log 2 4 + 1 2 log 2 2 + 1 4 log 2 4 = 3 2 . b. H [ Z ] = P ( Z = 1) log 2 1 P ( Z =1) + P ( Z = 2) log 2 1 P ( Z =2) + P ( Z = 3) log 2 1 P ( Z =3) = 1 3 log 2 3 + 1 3 log 2 3 + 1 3 log 2 3 ² 1 : 5850 Need: 30 1 : 5850 = 47 : 55 Answer: 48 bits c. H [ Z j X ] = X P ( X = b ) H [ Z j X = b ] = X b P ( X = b ) X a P ( Z = a j X = b ) log 2 1 P ( Z = a j X = b ) = P ( X = 5) ² 3 1 3 log 2 3 ³ + P ( X = 2) ² 3 1 3 log 2 3 ³ + P ( X = ± 3) ² 3 1 3 log 2 3 ³ = log 2 3 ² 1 : 5850 d. H [ Y j X;Z ] = X x;z P ( X = x;Z = z ) E [ Y j X = x;Z = z ] = = X x;z P ( X = x;Z = z ) X y P ( Y = y j X = x;Z = z ) log 2 1 P ( Y = y j X = x;Z = z ) = = X x 2f 2 ; 5 g z 2f 1 ; 2 ; 3 g 1 12 1 log 2 1 + X x = 3 z 2f 1 ; 2 ; 3 g 1 6 1 log 2 1 = 0 : e. The meaning of the answer for (c) is that since E [ Z j X ] = E [ Z ] , knowing X does not provide information about Z . BONUS. a. H [ X;Y ] = H [ X ] + H [ Y ] (since
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Unformatted text preview: X and Y are indpendent) = 2 & X x 2f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g 1 6 log 2 6 = 2 & log 2 6 5 : 1700 H [ X + Y ] = 2 & 1 36 log 2 36 + 2 & 2 36 log 2 36 2 + 2 & 3 36 log 2 36 3 + 2 & 4 36 log 2 36 4 + 2 & 5 36 log 2 36 5 + 6 36 log 2 36 6 3 : 2744 Answer: H [ X;Y ] H [ X + Y ] 1 : 8956 b. 300 & H [ X + Y ] 300 & 3 : 2744 = 982 : 3200 1...
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