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midterm_soln - University of Toronto Department of...

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University of Toronto Department of Electrical February 28, 2006 & Computer Engineering ECE1502S — Information Theory Midterm Test Solution 1 . ( Keeping Both Eyes Open ) We know that X and Y are independent and X and Z are independent. Does this mean that X is independent of the pair Y, Z ? The answer is no! For example, let Y and Z be chosen independently and uniformly at random from { 0 , 1 } , and let X = Y Z where denotes modulo-two addition. Then X is independent of Y and X is independent of Z , but I ( X ; Y, Z ) = 1 bit. 2 . ( Entropy Inequalities ) In general, we have H ( X 2 | X 1 ) 1 2 H ( X 1 , X 2 ) H ( X 1 ). To see that 2 H ( X 2 | X 1 ) H ( X 1 , X 2 ) observe that H ( X 1 , X 2 ) = H ( X 1 ) + H ( X 2 | X 1 ) (chain rule) = H ( X 2 ) + H ( X 2 | X 1 ) (by stationarity) H ( X 2 | X 1 ) + H ( X 2 | X 1 ) (conditioning does not increase entropy) = 2 H ( X 2 | X 1 ) Similarly, H ( X 1 , X 2 ) = H ( X 1 ) + H ( X 2 | X 1 ) (chain rule) H ( X 1 ) + H ( X 2 ) (conditioning does not increase entropy) = H ( X 1 ) + H ( X 1 ) (by stationarity) = 2 H ( X 1 ) . (More cleverly, we may observe that H ( X 1 , X 2 ) = H ( X 1 ) + H ( X 2 | X 1 ), where clearly H ( X 2 | X 1 ) H ( X 2 ) = H ( X 1 ). Thus the numbers H ( X 2 | X 1 ), 1 2 H ( X 1 , X 2 ) and H ( X 1 ) form a non-decreasing arithmetic sequence, and the result follows.) 3 . ( Lempel-Ziv Parsing ) ( a
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