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Unformatted text preview: EE 376B/Stat 376B Handout #29 Information Theory Tuesday, June 8, 2006 Prof. T. Cover Solutions to Homework Set #9 1. Horse race. Three horses run a race. A gambler offers 3for1 odds on each horse. These are fair odds under the assumption that all horses are equally likely to win the race. The true win probabilities are known to be p = ( p 1 ,p 2 ,p 3 ) = 1 2 , 1 4 , 1 4 ¶ . (1) Let b = ( b 1 ,b 2 ,b 3 ), b i ≥ 0, ∑ b i = 1, be the amount invested on each of the horses. The expected log wealth is thus W ( b ) = 3 X i =1 p i log 3 b i . (2) (a) Maximize this over b to find b * and W * . Thus the wealth achieved in repeated horse races should grow to infinity like 2 nW * with probability 1. (b) Show that if instead we put all of our money on horse 1, the most likely winner, we will eventually go broke with probability 1. Solution: Horse race. (a) The growth rate is given by W ( b ) = X i p i log b i o i = X i p i log 3 b i = X p i log 3 + X p i log p i X p i log p i b i = log 3 H...
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This note was uploaded on 06/10/2008 for the course ECE 376B taught by Professor Tomcover during the Spring '05 term at Stanford.
 Spring '05
 TomCover

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