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2535Decision Threshold Under Asymmetric Classification CostsA desk incurs cost 1 for a false positive and cost 5 for a false negative. Under a calibrated logistic probability p = P(Y=1|x), above what threshold should it predict class 1 to minimize expected cost?机器学习困难derivation未尝试面试订阅3346Closest Point to the Origin Above a Budget LineMinimize x 2+y 2 subject to x+y\ge 1. Find (x *,y *) and the optimal KKT multiplier for the inequality g(x,y)=1-x-y\le 0.数学中等derivation未尝试面试订阅3351Weighted Quadratic with Cheap x and Expensive yMinimize 1x 2+3y 2 subject to x+y\ge 4. Find (x *,y *) and the KKT multiplier.数学中等derivation未尝试面试订阅5271Minimum-Variance Weight 1Two risky assets have variances 0.04 and 0.09 and covariance 0.015. What weight on asset 1 gives the two-asset minimum-variance portfolio?金融与交易中等数值题未尝试面试订阅5278Variance Needed For One-Third Tangency WeightTwo uncorrelated risky assets have expected returns 0.10 and 0.14, risk-free rate 0.05, and asset 1 variance 0.05. What variance would asset 2 need so that the tangency portfolio puts exactly one-third of the risky allocation in asset 2?金融与交易困难数值题未尝试面试订阅5279Fee That Shrinks Tangency Weight To 20%Two uncorrelated risky assets have expected returns 0.06 and 0.09, risk-free rate 0.02, and variances 0.025 and 0.07. Suppose asset 2 carries an annual implementation fee c that reduces its expected return from 0.09 to 0.09-c. What fee c would make the tangency portfolio weight on asset 2 equal to 20%?金融与交易困难数值题未尝试面试订阅5280Variance Penalty For Equal Tangency WeightsTwo uncorrelated risky assets have expected returns 0.09 and 0.13, risk-free rate 0.04, and variances 0.035 and 0.10. Risk control adds a variance penalty x to asset 1, so its effective variance becomes 0.035 + x. What x would make the tangency portfolio split the risky allocation equally across the two assets?金融与交易困难数值题未尝试面试订阅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