h8_08 - INFORMATION THEORY First Quarter, Course 2010/2011...

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HOMEWORK #1 First Quarter, Course 2010/2011 Due Date: 29/10/2010 (friday, in-class) c ± Baltasar Beferull Lozano General Instructions and Comments The problems are not necessarily given in increasing order of difficulty. A final result that is correct does not guarantee the maximum grade in an exercise, in the same way as an incorrect final result does not imply a 0 grade in an exercise. You should explain clearly your arguments and show all the relevant steps in each of the problems. However, please, avoid redundant verbosity. The grade obtained in a problem will be based mainly in judging what is your level of understanding when solving the problem, as it is reflected from your writing. If any section of your solution of a problem is not legible, that section will be declared null (0 points). You are welcome to flag topics of confusion to you in any problem set; this will not lower your grade. You are expected to do yourself alone all the problems; I will assume so in making up the final exam. Problem 1. Understanding well Markovity Suppose the random variables A , B , C , D form a Markov chain A B C D . Then, (a) Is A B C ? (b) Is A C D ? (c) Is B C D ? (d) Is C B A ? (e) Is A ( B,C ) D ? (f) Is A B ( C,D ) ? Suppose now that the random variables A , B , C , D satisfy that: (i) A B C and (ii) B C D . Then, does it follow from these that A B C D ? Problem 2. Inequalities Label each of the following statements with =, or . Label a statement with = if equality always holds. Label a statement with or if strict inequality is possible. 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 ) (e) I ( X 1 ,X 2 ; Y 1 ,Y 2 ) vs. I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ), if p ( y 1 ,y 2 | x 1 ,x 2 ) = p ( y 1 | x 1 ) .p ( y 2 | x 2 ). (f)
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h8_08 - INFORMATION THEORY First Quarter, Course 2010/2011...

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