# hw2 - EE 376A Prof T Weissman Information Theory Thursday...

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EE 376A Information Theory Prof. T. Weissman Thursday, January 21, 2010 Homework Set #2 (Due: Thursday, January 28, 2010) 1. Prove that (a) Data processing decreases entropy: If Y = f ( X ) then H ( Y ) H ( X ). [ Hint: expand H ( f ( X ) , X ) in two different ways.] (b) Data processing on side information increases entropy: If Y = f ( X ) then H ( Z | X ) H ( Z | Y ). (c) Assume Y and Z are conditionally independent given X , denoted as Y X Z . In other words, P { Y = y | X = x, Z = z } = P { Y = y | X = x } for all x ∈ X , y Y , z ∈ Z . Prove that H ( Z | X ) H ( Z | Y ). 2. Entropy of a disjoint mixture. Let X 1 and 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 , . . . , n } . Let Θ be the result of a biased coin flip, i.e., P { Θ = 1 } = α and P { Θ = 0 } = 1 α . X 1 , X 2 and Θ are mutually independent. X = braceleftbigg X 1 , if Θ = 1 , X 2 , if Θ = 0 . (a) Find H ( X ) in terms of H ( X 1 ) and H ( X 2 ) and α. (b) Maximize over α to show that 2 H ( X ) 2 H ( X 1 ) + 2 H ( X 2 ) and interpret using the notion that 2 H ( X ) is the effective alphabet size.

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