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**Unformatted text preview: **Harvard SEAS ES250 – Information Theory Homework 3 (Due Date: Oct. 25 2007) 1. Let X ∼ p ( x ), x = 1 , 2 , ··· , m , denote the winner of a horse race. Suppose the odds o ( x ) are fair with respect to p ( x ), i.e., o ( x ) = 1 p ( x ) . Let b ( x ) be the amount bet on horse x , b ( x ) ≥ 0, ∑ m 1 b ( x ) = 1. Then the resulting wealth factor is S ( x ) = b ( x ) o ( x ), with probability p ( x ). (a) Find the expected wealth ES ( X ). (b) Find W * , the optimal growth rate of wealth. (c) Suppose Y = ‰ 1 , X = 1 or 2 , otherwise If this side information is available before the bet, how much does it increase the growth rate W * ? (d) Find I ( X ; Y ) 2. Many years ago in ancient St. Petersburg the following gambling proposition caused great conster- nation. For an entry fee of c units, a gambler receives a payoff of 2 k units with probability 2- k , k = 1 , 2 , ··· . (a) Show that the expected payoff for this game is infinite. For this reason, it was argued that c = ∞ was a ”fair” price to pay to play this game. Most people find this answer absurd. (b) Suppose that the gambler can buy a share of the game. For example, if he invests c/ 2 units in the game, he receives 1 2 a share and a return X/ 2, where Pr( X = 2 k ) = 2- k , k = 1...

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