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2379Variance Formula for an Antithetic Average 9If X and X' have the same variance sigma 2 and correlation rho, what is Var((X+X')/2)?数学中等derivation未尝试面试订阅2380Why a Stratified Estimator Is Unbiased 10Why is the weighted sum of unbiased within-stratum sample means unbiased for the overall expectation?数学困难derivation未尝试面试订阅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2383Law of Total Variance in a Regime Simulator 13What two pieces make up Var(X) under the law of total variance when a simulator first samples a regime Z and then samples X conditional on Z?数学中等derivation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2386Infer Regime Probability From Conditional Means 16A payoff has conditional mean 5 in a calm regime and -2 in a stress regime. The unconditional mean is 2.2. What calm-regime probability is implied?数学简单数值题未尝试免费2387Why Importance Sampling Stays Unbiased 17Why does reweighting by the likelihood ratio keep an importance-sampling estimator unbiased for the original expectation?数学中等derivation未尝试免费2388Why Rare Events Cause Huge Monte Carlo Noise 18Why can a rare-event payoff have an unstable Monte Carlo estimate even when most simulated paths look harmless?数学中等derivation未尝试免费2389Why Regime Averaging Can Beat Crude Mixing 19Why can separately estimating conditional means by regime and then averaging them outperform one crude pooled Monte Carlo run?数学困难derivation未尝试面试订阅2390Why Antithetics May Fail on a Highly Nonmonotone Payoff 20Why can antithetic pairing fail to reduce variance for a payoff that oscillates strongly across the state space?数学困难derivation未尝试面试订阅239195% Half-Width From Sample Volatility and Path Count 21A Monte Carlo estimator has sample standard deviation 5 across n=400 paths. Using the normal approximation, what is the 95% half-width?数学简单数值题未尝试免费2392Required Paths for a Target Standard Error 22A crude estimator has standard deviation 6. How many paths are needed to bring the standard error down to 0.3?数学简单数值题未尝试免费2393RMSE of an Unbiased Monte Carlo Estimate 23An unbiased estimator has standard deviation 4 over one path and uses n=64 independent paths. What is the RMSE of the sample mean?数学中等数值题未尝试免费2394Confidence Interval From Mean and Standard Error 24A Monte Carlo estimate is 12.0 with standard error 0.4. Using the normal approximation, what 95% confidence interval do you report?数学中等数值题未尝试免费2395Why Bias Still Matters Alongside Variance 25Why is a low-variance Monte Carlo estimator still problematic if it is biased?数学困难derivation未尝试面试订阅2396Variance of an Equal-Weight Correlated EnsembleFive base models each have prediction variance 4, and every pair of model predictions has correlation 0.25. If you average the five predictions equally, what is the ensemble variance?机器学习简单derivation未尝试免费2397Sample Size Crossover Between Two Model FamiliesModel A has excess test MSE 0.04 + 18/n, while model B has excess test MSE 0.16 + 4/n, where n is sample size. At what sample size do they tie?机器学习简单derivation未尝试免费2398Bias Budget Implied by a Variance ReductionA regularization change reduces a model's variance term from 0.30 to 0.11 while leaving irreducible noise unchanged. How much extra bias squared could you add before the total MSE stops improving?机器学习中等derivation未尝试免费2399Optimal Weight on a Noisy Unbiased ModelModel A is unbiased with variance 9. Model B has variance 1.44 and fixed bias 0.6. If you blend them as P w = wA + (1-w)B and treat their errors as independent, what weight w minimizes MSE?机器学习困难derivation未尝试面试订阅