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1955Large Buy-Side Imbalance Skew 10A quoting engine values upside fill opportunities much more than downside ones, so the optimal skew should be meaningfully positive. A market maker chooses a skew x in (-1,1) to maximize G(x) = 9 ln(1+x) + 3 ln(1-x). What skew is optimal?数学困难derivation未尝试免费1960Derive the Optimizer for an Exponential Asymmetry Objective 15For A>0 and B>0, derive the unique minimizer of J(x)=A e x + B e -2x .数学困难derivation未尝试免费2022Funding Buffer Gap From Two Leverage States 2The leverage can be either very low or quite high, and the desk wants the exact Jensen gap. A funding-buffer model uses phi(L)=1/(1+L). Suppose L takes values 0 and 3 with probabilities 1/2 and 1/2. Compute E[phi(L)] and phi(E[L]).数学简单数值题未尝试免费2025Reciprocal Funding Buffer Is Convex 5Let phi(L)=1/(1+L) on L>-1. State the Jensen inequality relation between E[phi(L)] and phi(E[L]).数学困难derivation未尝试面试订阅2035Funding Buffer Gap With Unequal Scenario Weights 15The high-leverage state is rarer, but still materially affects the convex average. A funding-buffer model uses phi(L)=1/(1+L). Suppose L takes values 1 and 4 with probabilities 1/4 and 3/4. Compute E[phi(L)] and phi(E[L]).数学困难数值题未尝试面试订阅2043Log Carry Gap From Two Scenarios 23The desk wants to see the exact concave Jensen gap, not just the inequality direction. A desk scores carry through psi(x)=ln(1+x). Suppose X takes values 0 and 3 with probabilities 1/2 and 1/2. Compute E[psi(X)] and psi(E[X]).数学中等数值题未尝试面试订阅2053When the Exponential Equation Is Safe for Newton 8If c>0 in exp(x)+cx=d, why is the Newton denominator never zero?数学中等derivation未尝试免费2067Second Two-Cashflow Yield Newton Step 22Another bond calibration step is computed with the same Newton machinery. Solve 6/(1+y) + 106/(1+y) 2 = 101 by one Newton step starting from y 0 = 0.04.数学中等数值题未尝试免费2101Recover the Missing Log Return From a Variance Fixing 6A variance-swap fixing uses RV = (252/n) * sum r i 2 with n=4. Three logged returns have already been recorded: [0.01, -0.02, 0.015]. What absolute missing return |r 4| would make the realized variance equal 0.04725?数理金融简单数值题未尝试免费2104Recover the Missing Log Return From a Variance Fixing 9A variance-swap fixing uses RV = (252/n) * sum r i 2 with n=4. Three logged returns have already been recorded: [0.012, 0.011, -0.009]. What absolute missing return |r 4| would make the realized variance equal 0.024885?数理金融简单数值题未尝试免费2105Recover the Missing Log Return From a Variance Fixing 10A variance-swap fixing uses RV = (252/n) * sum r i 2 with n=4. Three logged returns have already been recorded: [0.015, -0.005, -0.012]. What absolute missing return |r 4| would make the realized variance equal 0.045234?数理金融简单数值题未尝试免费2372Infer the Known Control Mean From the Adjusted Estimate 2A raw Monte Carlo estimator has sample mean Xbar=9. The control sample mean is Ybar=41, the desk uses b*=0.5, and the adjusted estimate Xbar-b(Ybar-mu Y) equals 8. What known control mean mu Y is implied?数学简单数值题未尝试免费2374Why the Control-Variate Estimator Is Unbiased 4Why is X-b(Y-E[Y]) unbiased for E[X] for any fixed constant b?数学中等derivation未尝试免费2375Why the Optimal Control Weight Equals Cov/Var 5Why does the variance-minimizing control weight satisfy b*=Cov(X,Y)/Var(Y)?数学困难derivation未尝试免费2376Paired-Difference Standard Error 6Two pricing engines are compared with common random numbers. The paired differences have sample standard deviation s D=2.8 across n=49 shared paths. What is the standard error of the estimated mean difference?数学简单数值题未尝试免费2377Required Shared Paths for a Half-Width Target 7A common-random-numbers comparison has paired-difference standard deviation s D=3.5. Using a 95% normal half-width target of 0.49, about how many shared paths are needed?数学中等数值题未尝试免费2378Why Common Random Numbers Help Comparisons 8Why can common random numbers reduce the variance of the difference between two estimators even if each estimator on its own is unchanged?数学中等derivation未尝试免费2382Remaining Variance Fraction Under Pairing 12Without common random numbers, two independent estimators have standard deviations 4 and 5, so the variance of their difference is 4 2+5 2. With pairing, the observed paired-difference standard deviation is 3. What fraction of the unpaired difference variance remains?数学简单数值题未尝试免费2384Why Regime Conditioning Can Reduce Variance 14Why can conditioning on a regime variable before sampling payoffs reduce Monte Carlo variance?数学困难derivation未尝试面试订阅2385When a Control Variate Can Make Things Worse 15Why can a badly chosen control coefficient increase variance instead of reducing it?数学困难derivation未尝试面试订阅