# hw3 - Mimic Theorem 8.6.5 in the text See also Chapter 12 9...

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ECE 5620 Spring 2011 Homework #3: Issued Feb. 16, Due Mar. 4 at 5 p.m. Complete all ten problems. 1. Problem 12 in Chapter 7 of the text. Omit the part about an intuitive reason. 2. Problem 19 in Chapter 7 of the text. The carrier pigeon channel can be viewed as a simple model of Internet communication. 3. Problem 20 in Chapter 7 of the text 4. Problem 23 in Chapter 7 of the text. You may do Problem 29 instead. 5. Problem 28 in Chapter 7 of the text. Omit part (b). 6. Problem 5 in Chapter 8 of the text. 7. “Most of the volume in a high-dimensional sphere is contained in a thin shell near the sphere’s surface.” Make this statement precise and prove it. 8. (a) Show that the geometric distribution with mean μ has the highest en- tropy of all discrete distributions on the nonnegative integers with mean μ . (b) Show that the exponential distribution with mean 1 has the highest differential entropy of all continuous distributions on [0 , ) with mean 1 . Hint:

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Unformatted text preview: Mimic Theorem 8.6.5 in the text. See also Chapter 12. 9. A Second Chance I Suppose that W is a random variable with alphabet W . Let ˆ W 1 and ˆ W 2 denote two guesses of W . Let P e denote the probability that both guesses are wrong P e = Pr( W 6 = ˆ W 1 and W 6 = ˆ W 2 ) . Since there is no point in repeating a guess, we assume that ˆ W 1 6 = ˆ W 2 with probability 1. (a) Show that H ( P e ) + P e log( |W| -2) + (1-P e ) ≥ H ( W | ˆ W 1 , ˆ W 2 ) . (b) Under what conditions does this inequality hold with equality? 10. A Second Chance II Consider a channel coding scenario in which the decoder gets two chances to correctly guess the transmitted message. We say that an error has occurred if both guesses are incorrect. Let C 2 denote the capacity of the channel under this setup, and let C denote the usual Shannon capacity. (a) Explain why C 2 ≥ C . (b) Does C 2 = C ? Prove or disprove....
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hw3 - Mimic Theorem 8.6.5 in the text See also Chapter 12 9...

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