10. Consider a gambler who wins a game with probability 0.4 and loses with probability 0.6....

 

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10. Consider a gambler who wins a game with probability 0.4 and loses with probability 0.6. When the bet size is $1, the gambler's wealth increases by $2 if he wins the game and decreases by $1 if he loses the game. The gambler plays the game repeatedly while betting p-fraction of his wealth in every game (0 ≤ p ≤ 1). Let Wn be the gambler's wealth after playing the nth game, and assume that W₁ = 1 (i.e., W₁ = 1 + 2p if he wins the first game, and W₁ 1 - p if he loses the first game). = 1 (a) Find E[W] and the long-run growth rate of the gambler's expected wealth, i.e., limnolog E[Wn]. (b) Find the optimal betting fraction p* that maximizes limn→∞log E[Wn]. (c) Find the long-run growth rate of the gambler's wealth (without expectation), i.e., limn→∞log Wn. (d) Find the optimal betting fraction p* that maximizes limn→∞log Wn. 10. Consider a gambler who wins a game with probability 0.4 and loses with probability 0.6. When the bet size is $1, the gambler's wealth increases by $2 if he wins the game and decreases by $1 if he loses the game. The gambler plays the game repeatedly while betting p-fraction of his wealth in every game (0 ≤ p ≤ 1). Let Wn be the gambler's wealth after playing the nth game, and assume that W₁ = 1 (i.e., W₁ = 1 + 2p if he wins the first game, and W₁ 1 - p if he loses the first game). = 1 (a) Find E[W] and the long-run growth rate of the gambler's expected wealth, i.e., limnolog E[Wn]. (b) Find the optimal betting fraction p* that maximizes limn→∞log E[Wn]. (c) Find the long-run growth rate of the gambler's wealth (without expectation), i.e., limn→∞log Wn. (d) Find the optimal betting fraction p* that maximizes limn→∞log Wn.

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