hw9 - b 1-b The odds are superfair(a There is a bet b that...

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EE 376B/Stat 376B Handout #24 Information Theory Thursday, June 1, 2006 Prof. T. Cover Due Tuesday, June 6, 2006 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. 2. Dutch book. Consider a horse race with m = 2 horses, X = 1 , 2 P = 1 2 , 1 2 odds (for one) = 10 , 30 bets =

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Unformatted text preview: b, 1-b. The odds are superfair. (a) There is a bet b that guarantees the same payoﬀ regardless of which horse wins. Such a bet is called a Dutch book . Find this b and the associated wealth factor S ( X ) . (b) What is the maximum growth rate of the wealth for the optimal choice of b ? Compare it to the growth rate for the Dutch book. 1 3. Investment growth rate. Let X = ( (1 ,a ) , with probability 1 / 2 (1 , 1 /a ) , with probability 1 / 2 , where a > 1. This vector X represents a stock market vector of cash vs. a hot stock. Let W ( b ,F ) = E log b t X and W * = max b W ( b ,F ) be the growth rate. (a) Find the log optimal portfolio b * . (b) Find the growth rate W * . (c) Find the asymptotic behavior of S n = n Y i =1 b t X i for all b . 2...
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hw9 - b 1-b The odds are superfair(a There is a bet b that...

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