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hw9sol - EE 376B/Stat 376B Information Theory Prof T Cover...

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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 3-for-1 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 ( p ) - D ( p || b ) log 3 - H ( p ) , with equality iff p = b . Hence b * = p = ( 1 2 , 1 4 , 1 4 ) and W * = log 3 - H ( 1 2 , 1 4 , 1 4 ) = 1 2
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