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hw1-2011 - ECE 563 Fall 2011 Homework I Issued August 30th...

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ECE 563, Fall 2011 Homework I Issued: August 30th, 2011 Due: September 13th, 2011 1. Entropy and Majorization. Entropy after Mixing Symbols . The interpretation of entropy as a measure of uncertainty suggests that “more uniform” distributions have larger entropy. For two distributions P and Q on X we call P “more uniform” than Q , symbolically P > Q , if for the non-increasing ordering , | X |= n , of their probabilities, it holds that for every Show that P > Q implies H( P ) H( Q ). As a special case, consider what happens to the entropy when two outcomes (say, a and b ) of two random variables are “mixed”, i.e., the probability of symbols a and b are both replaced by the average (p(a)+p(b))/2. 2. Mutual information of heads and tails. a. Consider a fair coin flip. What is the mutual information between the top side and the bottom side of the coin? b. A 6-sided fair die is rolled. What is the mutual information between the top side and the bottom side?
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