hw2 - EE595A Introduction to Information Theory Winter 2004...

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EE595A Introduction to Information Theory University of Washington Winter 2004 Dept. of Electrical Engineering Handout 4: Problem Set 2 Prof: Jeff A. Bilmes <[email protected]> Lecture 6, Jan 26, 2004 DUE: Wednesday, Feb 4th, 2004, in class Book Problems Problem 1: Let S be a random variable with a binomial distribution with parameters n and p so that P ( S = i ) = ± n i ² p i (1 - p ) n - i show that the most likely value of S is the value <np> , where <np> is the integer closest to np . Hint: use the fact that P ( S = i + 1) P ( S = i ) = ( n - i ) p ( i + 1)(1 - p ) and consider the cases when i < np , and i > np . Problem 2: Consider a sphere of radius r in N dimensions. Show that the fraction of the volume of the sphere that lies at values of the radius between r - ± and r , where 0 < ± < r , is: f = 1 - ³ 1 - ± r ´ N . Using matlab, evaluate f for the cases N = 2 ,N = 10 , and N = 1000 with a) ±/r = 0 . 01 ; and b) ±/r = 0 . 5 . Notice that points that are uniformly distributed in a sphere in N dimensions are very likely to be in a thin shell near the surface when N is large. Note that you might find useful the volume of a hypersphere of radius
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hw2 - EE595A Introduction to Information Theory Winter 2004...

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