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Unformatted text preview: EE 478 Handout #3 Multiple User Information Theory Thursday, September 25, 2008 Homework Set #1 Due: Thursday, October 2, 2008. 1. Inequalities. Label each of the following statements with =, ≤ , or ≥ . Justify each answer. (a) H ( X | Z ) vs. H ( X | Y ) + H ( Y | Z ). (b) h ( X + Y ) vs. h ( X ), if X and Y are independent continuous random variables. (c) I ( g ( X ); Y ) vs. I ( X ; Y ). (d) I ( Y ; Z | X ) vs. I ( Y ; Z ), if p ( x,y,z ) = p ( x ) p ( y ) p ( z | x,y ). 2. Hadamard inequality. Let Y n ∼ N(0 ,K ). (a) Show that h ( Y n ) ≤ 1 2 log (2 πe ) n n Y i =1 K ii ! . (b) Use the result of part (a) to prove the Hadamard inequality det( K ) ≤ n Y i =1 K ii for all positive semidefinite K . 3. Csisz´ ar sum formula. Let X n and Y n be two random vectors with arbitrary joint probability distribution. Show that n X i =1 I ( X n i +1 ; Y i | Y i- 1 ) = n X i =1 I ( Y i- 1 ; X i | X n i +1 ) for 1 ≤ i ≤ n , where X n +1 ,Y = ∅ . As we shall see later, this inequality is useful in....
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This note was uploaded on 10/24/2011 for the course ELECTRICAL ECE 571 taught by Professor Kelly during the Spring '11 term at University of Illinois, Urbana Champaign.
- Spring '11