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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4891Capital Aggregation Diagnostic 1A CRO wants one firm-wide capital number for two desks whose books can offset each other. Which coherence property should you check first if you do not want the merged capital to exceed the sum of standalone capitals?数理金融中等essay未尝试面试订阅4892Capital Aggregation Diagnostic 2Treasury injects a deterministic cash buffer into every scenario of a loss distribution. Which coherence property tells you how the required capital should move after that cash injection?数理金融中等essay未尝试面试订阅4893Capital Aggregation Diagnostic 3A PM doubles every position in a book and wants to know whether capital should double too. Which coherence property is the first one to inspect?数理金融中等essay未尝试面试订阅4894Capital Aggregation Diagnostic 4Book X never loses more than book Y in any scenario. Which coherence axiom says X should not require more capital than Y?数理金融中等essay未尝试面试订阅4895Capital Aggregation Diagnostic 5Two books are each acceptable on their own under a coherent capital rule. Why should a convex mixture of the two also be acceptable?数理金融中等essay未尝试面试订阅4896Translation-Invariant Capital Recovery 6A coherent capital rule reports rho(L-h)=4.6 after a deterministic hedge h=1.2 is added in every scenario. What was rho(L), and how much extra deterministic cash would still be needed to make the hedged position acceptable?数理金融简单数值题未尝试面试订阅4897Translation-Invariant Capital Recovery 7A book has coherent capital rho(L)=7.1. Treasury wants the post-cash capital to be 2.3. How large a deterministic cash injection is needed, and how much total deterministic cash would make the original book acceptable?数理金融简单数值题未尝试面试订阅