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1499Exponential Tail Integral 4Evaluate integral 0 inf x 2 e (-2x) dx.数学中等derivation未尝试免费1501Quote Impact Multiplier 1Use a second-order Taylor approximation around 0 to estimate exp(2x)/(1+1x) at x=1/20.数学简单数值题未尝试免费1503Inventory Skew Multiplier 3Use a second-order Taylor approximation around 0 to estimate exp(1x)/(1+-1x) at x=1/25.数学中等derivation未尝试免费1504Spread Fade Multiplier 4Use a second-order Taylor approximation around 0 to estimate exp(-2x)/(1+1x) at x=1/40.数学中等数值题未尝试免费1506Volatility Rescaling Neutralizer 1Choose b so that sqrt(1+6x) * exp(bx) has no linear term in its Taylor expansion around 0.数学简单数值题未尝试免费1511Local Log-Carry Curvature 1Compute the x 2 coefficient in ln(1+5x) - ln(1+2x) - (3)x.数学简单数值题未尝试免费1512Cross-Asset Log Ratio Curvature 2Compute the x 2 coefficient in ln(1+4x) - ln(1+-1x) - (5)x.数学简单数值题未尝试免费1516Levered Book Approximation 1Use a second-order Taylor approximation around 0 to estimate (1+2x) (3/2) * (1+-1x) (-1) at x=1/20.数学简单数值题未尝试免费1517Convex Funding Approximation 2Use a second-order Taylor approximation around 0 to estimate (1+1x) (1/2) * (1+3x) (-1) at x=1/50.数学中等数值题未尝试免费1518Skewed Ratio Approximation 3Use a second-order Taylor approximation around 0 to estimate (1+4x) (-1) * (1+-2x) (1/2) at x=1/25.数学中等derivation未尝试免费1519Adjusted Gross Multiplier 4Use a second-order Taylor approximation around 0 to estimate (1+3x) (2) * (1+1x) (-1) at x=1/40.数学困难derivation未尝试面试订阅1520Local Hedge Multiplier 5Use a second-order Taylor approximation around 0 to estimate (1+-2x) (1/2) * (1+5x) (1/2) at x=1/100.数学困难derivation未尝试面试订阅1521Quadratic Stress Calibration 1Choose c so that exp(2x + c x 2) - 1 - 2x has quadratic coefficient 5.数学简单数值题未尝试免费1523Asymmetric Buffer Calibration 3Choose c so that exp(-1x + c x 2) - 1 - -1x has quadratic coefficient 4.数学中等数值题未尝试免费1757Recovering Signal-to-Noise From Two Attenuated SlopesA latent factor X* is measured twice: X1 = X* + e1 and X2 = X* + e2, where e1 and e2 are independent classical errors with the same variance and are independent of X*. Regressing Y on X1 alone gives slope 0.80. Regressing Y on the average (X1+X2)/2 gives slope 1.00. Under the same true structural slope beta, what is Var(X*)/Var(e)?统计简单derivation未尝试免费1767Weak-IV Risk From the First Stage AloneA proposed instrument shifts treatment by 0.02 with standard error 0.015 in the first stage. Even if the exclusion story sounds plausible, what first-stage F-statistic do you get, and what is the main identification concern?统计中等derivation未尝试面试订阅1829MA(1) Lag-1 Correlation 4A microstructure noise model uses Y t = e t + 1 e (t-1). What is its lag-1 autocorrelation rho(1)?统计中等derivation未尝试面试订阅1946Log Utility Versus Impact Budget 1A signal is strong enough that the PM scores larger size through a log benefit but pays quadratic impact. The desk chooses a scalar exposure x > -1 to maximize F(x) = 3 ln(1+x) - 2 x 2. What exposure x maximizes F?数学简单数值题未尝试免费1947Log Utility Versus Impact Budget 2A desk chooses a participation tilt with diminishing marginal benefit and quadratic implementation drag. The desk chooses a scalar exposure x > -1 to maximize F(x) = 4 ln(1+x) - 1 x 2. What exposure x maximizes F?数学简单数值题未尝试免费1948Log Utility Versus Impact Budget 3A portfolio manager sizes one smooth alpha sleeve with a log reward and a quadratic risk surcharge. The desk chooses a scalar exposure x > -1 to maximize F(x) = 12 ln(1+x) - 1 x 2. What exposure x maximizes F?数学中等数值题未尝试免费