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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1440Square-Root Remainder 5Compute lim (x->0) [sqrt(1 + 3x + -4x 2) - 1 - (3/2)x] / x 2.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1444Trigonometric Asymptotic 4Compute lim (x->0) [arctan(5x) - 5x] / x 3.数学困难数值题未尝试面试订阅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1448Mixed-Expansion Limit 3Compute lim (x->0) [ln(1+2x) - 2 arctan(x)] / x 2.数学中等数值题未尝试免费1449Mixed-Expansion Limit 4Compute lim (x->0) [e x cos(x) - 1 - x] / x 2.数学中等数值题未尝试免费1450Mixed-Expansion Limit 5Compute lim (x->0) [ln(cosh(2x))] / x 2.数学困难数值题未尝试面试订阅1526Determinant After a Rank-One Overlay 1An invertible matrix A has det(A) = 12. For vectors u and v, suppose v T A (-1) u = 1/2. What is det(A + u v T)?数学简单数值题未尝试免费1527Stress-Adjusted Determinant 2An invertible matrix A has det(A) = 5. For vectors u and v, suppose v T A (-1) u = -3/5. What is det(A + u v T)?数学简单数值题未尝试免费1531Residual-Space Projection Rank 1P is a projection on R 8 with rank 3. What is rank(I - P)?数学简单数值题未尝试免费1541Sherman-Morrison Vector Update A 1Suppose A is invertible, A (-1)b = [3, -1] T, A (-1)u = [1, 2] T, v T A (-1)b = 2, and v T A (-1)u = 1/2. What is (A + u v T) (-1) b?数学简单数值题未尝试免费1547Broadcast Matrix Determinant 2What is det(5 I + -1 11 T) as an n=3 dimensional matrix?数学简单数值题未尝试免费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1617Correlation from Covariance 1Two assets have variances 9 and 16, and covariance 6. What is their correlation?数学中等derivation未尝试免费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1778Lasso Threshold Calibration 3In an orthonormal lasso update, a coordinate has score z = 2.6 and penalty lambda = 1.1. What coefficient results after soft-thresholding?统计中等数值题未尝试免费1802Measurement Noise Effect 2A latent stationary signal Y t has gamma(0) = 8 and gamma(1) = 3. You observe X t = Y t + eta t, where eta t is iid noise with variance 2 independent of Y t. What are gamma X(0) and gamma X(1)?统计中等derivation未尝试免费