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1962Balanced Exponential Penalties 17A desk minimizes J(x)=6 e x + 3 e -2x . What x is optimal?数学中等derivation未尝试免费2679Why Hundreds of Stocks Do Not Mean Hundreds of Independent LabelsWhy does a daily cross-sectional equity sample with hundreds of names still provide much less information than its row count suggests?机器学习中等essay未尝试面试订阅5903Capping the Single-Bet DrawdownYou bet a fraction f of wealth on an even-money coin with win probability p=0.65, but a risk rule forbids any single losing bet from cutting your wealth by more than 20\%. What fraction should you bet, and for which win probabilities p does this drawdown rule actually constrain you below the Kelly fraction?概率简单数值题未尝试免费5905Kelly with a Cash-Reserve FloorYou may stake on an even-money coin with win probability p=0.8, but a liquidity rule requires you to keep at least half your total wealth in untouched cash at all times, so the staked fraction satisfies f\le 0.5. Set up the constrained maximization of expected log-growth, use the KKT conditions to determine the optimal stake, and state whether the reserve constraint binds.概率困难数值题未尝试面试订阅