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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1664Weight on the Noisier Signal to Hit a Variance TargetTwo independent unbiased estimators T1 and T2 have variances 1 and 4. In the blend W = wT2 + (1-w)T1, what is the smallest positive w that makes Var(W) exactly 0.9?统计简单derivation未尝试免费1665Interval Where a Fixed Benchmark Beats a Noisy Unbiased SignalA noisy unbiased signal X satisfies X ~ N(theta, 0.25). A fallback benchmark always reports 1.2. For what values of theta does the fixed benchmark have lower MSE than X?统计简单derivation未尝试免费1666Exact Bootstrap Variance of a Two-Point MeanA sample is a,b . In the nonparametric bootstrap, a resample of size 2 is drawn with replacement. Derive the variance of the bootstrap sample mean.统计简单derivation未尝试免费1667Bootstrap Bias of the Sample Maximum on a Tiny SampleA sample is 1,4 . Under the nonparametric bootstrap with resample size 2, compute the bootstrap expectation of the sample maximum and the resulting bootstrap bias estimate for the original maximum statistic.统计简单derivation未尝试免费1668Bootstrap Variance of a Bernoulli Mean From the Plug-In LawA sample of size n has empirical success rate p hat. Under the nonparametric bootstrap, what is the variance of the resampled sample mean conditional on the observed data?统计简单derivation未尝试免费1669When the Bootstrap Median Is Forced to Equal One of Two ValuesA sample is 0,0,10 . Under the nonparametric bootstrap with resample size 3, what values can the bootstrap median take, and what condition determines which one occurs?统计中等derivation未尝试面试订阅1670m-out-of-n Bootstrap Variance Scaling for a MeanIf a statistic is the sample mean and you use an m-out-of-n nonparametric bootstrap instead of an n-out-of-n bootstrap, how does the conditional variance of the resampled mean scale with m?统计简单derivation未尝试免费1671Why the Naive Bootstrap Misses a BoundaryAn estimator is constrained to be nonnegative and lands exactly at 0 on the observed sample. Why can the naive nonparametric bootstrap badly misrepresent uncertainty near that boundary?统计简单derivation未尝试免费1672Pairs Bootstrap Versus Residual Bootstrap in Heteroskedastic DataWhy can a residual bootstrap be invalid for regression under heteroskedasticity while a pairs bootstrap remains defensible?统计中等derivation未尝试面试订阅1673When the Bootstrap Unit Should Be a Cluster, Not a RowWhy should a practitioner resample accounts, users, or securities rather than individual rows when observations inside each group share latent shocks?统计中等essay未尝试面试订阅