# hw0 - EE 376A Information Theory Prof T Weissman Handout#1...

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EE 376A Handout #1 Information Theory Monday, January 04, 2010 Prof. T. Weissman NOT due Homework Set #0 Note: HW0 has NO effect on your grade of EE376A. There is no requirement to hand it in. Those are warm-up exercises in probability. However, we would like to hear your feedbacks. If you have some cute problems in probability, please share with us as well. 1. ( Independence ) (a) Let Z be a bernoulli random variables (r.v.) with parameter 1/2, (Bern(1/2)), i.e. P ( Z = 0) = 1 / 2, P ( Z = 1) = 1 / 2. Let X be a bernoulli random variables with parameter p and X is independent of Z . Show that the modulo 2 sum of X and Z is independent of X for any p value. (b) Let Z 1 ,...,Z n be independent Bern(1/2) r.v’s. Suppose ( X 1 ,...,X n ) is a random bi- nary sequence and independent of ( Z 1 ,...,Z n ). Let Y i = X i + Z i mod 2. What is the distribution of ( Y 1 ,...,Y n )? (c) Consider a more general case. Let Z be uniformly distributed over { 0 , 1 ,...,M - 1 } . X is independent of Z , with arbitrary probability mass function over { 0 , 1 ,...,M - 1 } . Show that the modulo M sum of X and Z is independent of X .

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• Fall '09
• Probability theory, Bernoulli random variables, Bern

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