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1438Sequence Limit From a Square RootCompute lim n->∞ n[(1+2/n) (1/2) - 1].数学中等数值题未尝试免费1439Logarithmic Sequence CorrectionCompute lim n->∞ n 2 [ln(1+1/n) - 1/n].数学困难数值题未尝试面试订阅1442Exponential Minus CosineCompute lim x->0 [e x - cos x - x] / x 2.数学中等数值题未尝试免费1443Arcsine Cubic RemainderCompute lim x->0 [arcsin x - x] / x 3.数学中等数值题未尝试免费1445Product-Log CancellationCompute lim x->0 [ (1+x)ln(1+x) - x ] / x 2.数学困难数值题未尝试面试订阅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1545Sherman-Morrison Failure TestWhy does the Sherman-Morrison formula break down exactly when 1 + v T A (-1)u = 0?数学困难数值题未尝试面试订阅1549Ridge-Plus-Ones Determinant 4What is det(4 I + 1/2 11 T) as an n=5 dimensional matrix?数学中等数值题未尝试免费1554Ridge Plus Crowd Mode 4A = 4I + 1/211 T in dimension n=6. What are its eigenvalues?数学中等derivation未尝试免费1601Why Every Covariance Matrix Is PSDWhy must every valid covariance matrix be positive semidefinite?数学简单essay未尝试免费1608Minimum-Variance Hedge Ratio 1You hedge X with h units of Y and want to minimize Var(X - hY). If Cov(X, Y) = 8 and Var(Y) = 4, what is the optimal hedge ratio h?数学中等derivation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1615Signed-Factor Covariance Entry 2Under a one-factor model X = bF + epsilon with factor variance 5, asset loadings (2, -1), and idiosyncratic variances (1, 4), 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1631Recovering Latent Regime Size from Second and Fourth MomentsA stylized one-period microstructure model writes the observed shock as Y = S a + \varepsilon, where S takes values +1 and -1 with equal probability, a>0 is an unknown regime magnitude, and \varepsilon \sim N(0, 2) is independent noise. From data, the empirical second moment is m 2 = 5 and the empirical fourth moment is m 4 = 43. Use the method of moments to estimate a and 2.统计困难derivation未尝试面试订阅1635Estimating a Three-Point Shock Model from Even MomentsA stylized inventory-shock model assumes the one-step PnL jump X takes values -a, 0, and +a with probabilities p/2, 1-p, and p/2, respectively, where a>0 is unknown. From data, the empirical second moment is 2 and the empirical fourth moment is 10. Use the method of moments to estimate p and a.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1674Why the Percentile Interval Can Drift Off-CenterWhy can a percentile bootstrap interval end up noticeably off-center relative to the original estimate when the statistic is skewed?统计中等essay未尝试面试订阅