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Unformatted text preview: to see a sequence of independent, equally likely 0’s and 1’s. Problem 7: Let X , Y and Z be three discrete random variables. Using Jensen’s inequality, or otherwise (e.g., the chain rule), show that I ( X,Y ; Z ) ≥ I ( Y ; Z ). When does the equality hold? Problem 8: Let X 1 , X 2 be discrete random variables drawn according to probability mass functions p 1 ( · ) and p 2 ( · ) over the respective alphabets X 1 = { 1 , 2 ,... ,m } and X 2 = { m + 1 , 2 ,... ,n } . Let X = b X 1 with probability α X 2 with probability 1α 1 Find H ( X ) in terms of H ( X 1 ) and H ( X 2 ) and α . Hint: de±ne a function of X as : γ = f ( X ) = b 1 when X = X 1 2 when X = X 2 and use H ( X,γ ). 2...
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 Fall '08
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 Probability theory, GEORGIA INSTITUTE OF TECHNOLOGY School of Electrical and Computer Engineering ECE

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