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5904Kelly Exceeds Full InvestmentA favorable bet has limited downside: staking a fraction f of wealth, you gain the full amount f with probability p=0.7 but lose only half the stake, 0.5f, with probability 0.3. (a) Find the growth-optimal fraction f *. (b) If you cannot borrow (so f\le 1, i.e. you can stake at most all your wealth), what fraction do you actually bet?概率简单数值题未尝试免费5907Kelly with a Proportional Trading CostOn an even-money coin with win probability p, each round you pay a proportional cost c on the amount staked, regardless of the outcome. So staking fraction f, a win multiplies wealth by 1+f(1-c) and a loss by 1-f(1+c). Derive the growth-optimal fraction f * in terms of p and c, evaluate it for p=0.6,\ c=0.05, and find the cost level at which the optimal stake drops to zero.概率困难数值题未尝试面试订阅