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4866Infer Antithetic Correlation From Variance Reduction 1A Monte Carlo desk averages each payoff with its antithetic partner. The variance of the antithetic average is observed to be 35% of the crude single-path variance. If the paired payoffs have equal variance, what correlation rho between the two payoffs is implied?数理金融简单数值题未尝试面试订阅4871Infer Covariance From Optimal Control Weight 6A control variate Y has sample variance Var(Y)=25. The optimal control coefficient is estimated as b*=0.8 in the estimator X - b*(Y-E[Y]). What Cov(X,Y) is implied?数理金融中等数值题未尝试面试订阅4872Infer Covariance From Optimal Control Weight 7A control variate Y has sample variance Var(Y)=4. The desk estimates the optimal coefficient b*=1.5 in X - b*(Y-E[Y]). What Cov(X,Y) is implied?数理金融中等数值题未尝试面试订阅4873Infer Covariance From Optimal Control Weight 8A control variate Y has sample variance Var(Y)=9. The estimated optimal control coefficient is b*=-0.6. What Cov(X,Y) is implied?数理金融中等数值题未尝试面试订阅4874Infer Known Control Mean From Adjusted Estimate 9A raw Monte Carlo estimator has sample mean Xbar=12. The control sample mean is Ybar=103, the desk uses b*=0.5, and the adjusted estimate Xbar - b*(Ybar-mu Y) equals 10.5. What known control mean mu Y is implied?数理金融中等数值题未尝试面试订阅4876Paired-Difference Standard Error 11Two pricing algorithms are compared with common random numbers. The paired difference sample has standard deviation s D=3 across n=100 shared paths. What is the standard error of the estimated mean difference?数理金融中等数值题未尝试面试订阅4877Paired-Difference Standard Error 12A common-random-numbers comparison produces paired differences with s D=2.4 over n=64 paths. What is the standard error of the sample mean difference?数理金融中等数值题未尝试面试订阅4878Paired-Difference Standard Error 13A common-random-numbers run yields paired differences with s D=5 over n=25 paths. What is the standard error of the mean difference?数理金融中等数值题未尝试面试订阅4880Variance Improvement From Pairing 15Without common random numbers, two independent estimators have standard deviations 4 and 5, so the variance of their difference would be 4 2+5 2. With pairing, the observed paired-difference standard deviation is 3. What fraction of the unpaired difference variance remains?数理金融中等数值题未尝试面试订阅4881Infer Regime Probability From Conditional Means 16A payoff has conditional mean 4 in a calm regime and -3 in a stress regime. The overall expected payoff is 1.9. What probability of the calm regime is implied?数理金融困难数值题未尝试面试订阅4882Infer Missing Stress Mean From Tower Equation 17A payoff has conditional mean 5 in a calm regime and unknown mean m in a stress regime. The calm regime probability is 0.6 and the overall expected payoff is 2.6. What is m?数理金融困难数值题未尝试面试订阅4883Infer Stress Probability From Conditional Means 18A payoff has conditional mean 2 in a normal regime and -4 in a stress regime. The overall expected payoff is 0.8. What probability of the stress regime is implied?数理金融困难数值题未尝试面试订阅4884Infer Missing Mid-Regime Mean In a Three-State Mixture 19A payoff has regime probabilities 0.5, 0.3, and 0.2. The conditional means are 1, m, and -2, and the overall expected payoff is 0.7. What is m?数理金融困难数值题未尝试面试订阅4885Infer Regime Exercise Probability From Overall Expectation 20In regime A, which occurs with probability 0.4, an option pays 10 if exercised. In regime B, which occurs with probability 0.6, it pays 4 if exercised, and the regime-B exercise probability is 0.25. The overall expected payoff is 2.6. What exercise probability in regime A is implied?数理金融困难数值题未尝试面试订阅4886Infer Second-Stratum Volatility From Target Neyman Share 21Two strata have population weights N1=0.6 and N2=0.4. Their standard deviations are sigma1=2 and sigma2=unknown. Under equal-cost Neyman allocation, the desk wants stratum 2 to receive 50% of the samples. What sigma2 is implied?数理金融中等数值题未尝试面试订阅4887Updated Neyman Share After a Volatility Change 22Two equal-size strata each have population weight 0.5. After a shock, sigma1 rises to 2 while sigma2 stays at 1.5. Under equal-cost Neyman allocation, what sample share should stratum 1 now receive?数理金融中等数值题未尝试面试订阅4888Infer Variance Ratio From Equal Neyman Shares 23Two strata have population weights N1=0.7 and N2=0.3. Under equal-cost Neyman allocation they end up with equal sample shares. What ratio sigma2/sigma1 is implied?数理金融中等数值题未尝试面试订阅4889Three-Stratum Neyman Share 24Three strata have population weights 0.5, 0.3, and 0.2, with standard deviations 1, 2, and 3. Under equal-cost Neyman allocation, what sample share should the third stratum receive?数理金融中等数值题未尝试面试订阅4890Updated Small-Stratum Share After Tail Risk Doubles 25Two strata have population weights 0.8 and 0.2. The first stratum has sigma1=1. The second stratum's standard deviation jumps from 4 to 8. Under equal-cost Neyman allocation, what sample share should the second stratum now receive?数理金融中等数值题未尝试面试订阅