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1543Sherman-Morrison Vector Update C 3Suppose A is invertible, A (-1)b = [5, 0] T, A (-1)u = [2, 1] T, v T A (-1)b = 1, and v T A (-1)u = -1/2. What is (A + u v T) (-1) b?数学中等essay未尝试免费1607Why Sample Covariance Can Be Rank DeficientWhy can a sample covariance matrix become rank deficient when the number of observations is smaller than the number of assets?数学中等essay未尝试免费1614One-Factor Covariance Entry 1Under a one-factor model X = bF + epsilon with factor variance 3, asset loadings (1, 2), and idiosyncratic variances (4, 9), what are Var(X 1), Var(X 2), and Cov(X 1, X 2)?数学困难derivation未尝试面试订阅1618Negative Correlation Extraction 2Two assets have variances 4 and 25, and covariance -6. What is their correlation?数学中等derivation未尝试免费1619Why Large Condition Number Destabilizes WeightsWhy does a covariance matrix with a very large condition number make optimized portfolio weights unstable?数学困难essay未尝试面试订阅1626Recovering a Three-Point Support Scale From Two MomentsA random variable takes values 0, a, and 3a with probabilities 1-2p, p, and p. If the empirical first two raw moments are m 1 and m 2, solve for a and p by method of moments.统计简单derivation未尝试免费1633Two-Rate Latency Mixture With Known Mixing WeightA latency variable is a 50-50 mixture of two exponential laws with rates \lambda 1 and \lambda 2. The first two raw moments are m 1 and m 2. Write the two equations that method of moments imposes on (\lambda 1,\lambda 2).统计中等derivation未尝试面试订阅1651Bias Budget for a Faster ProxyA slow benchmark estimator U is unbiased with variance 0.64. A faster proxy P has variance 0.25 but constant bias b. What is the largest absolute bias |b| for which P still has smaller MSE than U?统计中等derivation未尝试面试订阅1652Optimal Shrink Toward a Desk AnchorA desk observes X ~ N(theta, 9) and reports delta c = cX + (1-c)4. At the specific parameter value theta = 5, what choice of c minimizes MSE, and what is the minimum MSE?统计中等derivation未尝试面试订阅1653A Smoothed Bernoulli Estimator vs the Sample ProportionLet X\sim Binomial (10,p) and consider the estimator = X+1 12 for p. At the parameter value p=0.2, compute the bias, variance, and MSE of , and compare its MSE with the usual sample proportion p = X/10.统计中等derivation未尝试面试订阅1654When an Adjusted Signal Beats the Raw EstimateSuppose X ~ N(theta, 1). A desk uses the adjusted estimator delta = 0.75X + 0.5 instead of the raw signal X. For what values of theta does delta have smaller MSE than X?统计中等derivation未尝试面试订阅1655Stale Low-Noise Estimate vs Fresh Noisy EstimateA stale estimator S for today's parameter has variance 0.04 but incurs bias Delta because the regime has moved. A fresh estimator F is unbiased for today's parameter but has variance 0.25. What is the largest |Delta| for which S still has lower MSE than F?统计中等derivation未尝试面试订阅1656Sample Size Needed for a 40% MSE CutThe sample mean of n i.i.d. observations with variance sigma 2 has MSE sigma 2/n. A desk currently uses n = 10 observations. What total sample size is needed so the MSE is at most 60% of the current MSE?统计简单derivation未尝试免费1657Optimal Haircut on a Multiplicative Vol SignalAn estimator A is unbiased for theta and satisfies Var(A) = 0.3 theta 2. A risk team reports delta c = cA instead. Find the value of c that minimizes MSE, and give the minimum MSE as a multiple of theta 2.统计困难derivation未尝试面试订阅1658Best Blend of Two Correlated Unbiased SignalsTwo unbiased estimators of the same parameter have variances 9 and 4, and their correlation is 0.5. For T(a) = aT1 + (1-a)T2, what value of a minimizes variance, and what is the resulting minimum variance?统计简单derivation未尝试免费1659Bias Allowed for a Lower-Variance Regularized EstimateAn unbiased estimator U has variance 0.06. A regularized estimator R has variance 0.03 and constant bias b. What is the largest absolute bias |b| for which R still has smaller MSE than U?统计中等derivation未尝试面试订阅1660Norm Implied by a 25% Positive-Part James-Stein ShrinkIn dimension p = 4 with unit noise variance, the positive-part James-Stein shrinkage factor is 0.75 for an observed vector x. What value of ||x|| 2 produced that factor?统计中等derivation未尝试面试订阅1661Anchor-Based Shrinkage Crossover IntervalSuppose Xbar ~ N(theta, 0.16). A desk uses delta = 0.6Xbar + 0.8. For what values of theta does delta have lower MSE than Xbar?统计中等derivation未尝试面试订阅1662When Smaller Variance Beats a Vanishing BiasEstimator A is asymptotically unbiased with asymptotic MSE 1/n. Estimator B has asymptotic variance 0.7/n and asymptotic bias 1/n. From what smallest integer n onward does B have lower asymptotic MSE than A?统计简单derivation未尝试免费1663Sample Size for a Stable Variance EstimatorUnder a normal model, the unbiased sample variance satisfies Var(S 2) = 2 sigma 4 / (n-1). What is the smallest n for which the standard deviation of S 2 is at most 0.5 sigma 2?统计简单derivation未尝试免费