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1868RW Versus MR Diagnosis 3A five-day variance ratio comes in well below 1. What does that suggest about serial dependence in returns?统计中等essay未尝试面试订阅1869RW Versus MR Diagnosis 4If a spread's conditional mean always equals today's level, regardless of horizon, which model is the closer description?统计中等derivation未尝试面试订阅1870RW Versus MR Diagnosis 5A desk notices that expected residual carry from holding a spread for longer horizons quickly saturates instead of growing linearly forever. Is that more consistent with random walk or mean reversion?统计困难derivation未尝试面试订阅1954Symmetric Flow Implies Flat Skew 9If a market maker maximizes G(x)=a ln(1+x)+a ln(1-x) on x in (-1,1), what skew is optimal?数学中等derivation未尝试免费1955Large Buy-Side Imbalance Skew 10A quoting engine values upside fill opportunities much more than downside ones, so the optimal skew should be meaningfully positive. A market maker chooses a skew x in (-1,1) to maximize G(x) = 9 ln(1+x) + 3 ln(1-x). What skew is optimal?数学困难derivation未尝试免费1959Large Cushion Interior Minimum 14The reciprocal term is strong enough that the optimum sits well away from the boundary. Minimize H(x) = 1 x 2 + 36/(1+x) over x > -1.数学困难derivation未尝试免费1960Derive the Optimizer for an Exponential Asymmetry Objective 15For A>0 and B>0, derive the unique minimizer of J(x)=A e x + B e -2x .数学困难derivation未尝试免费1963How the Exponential Optimum Moves With the Back-End Penalty 18For J(x)=A e x + B e -2x , suppose A stays fixed and B is quadrupled. How does the optimizer x* move?数学中等derivation未尝试免费1965Derive the Optimizer for a Linear-Softplus Objective 20For 0<m<n, derive the unique maximizer of K(x)=m x - n ln(1+e x).数学困难derivation未尝试免费1970Maximum Value at the Interior Utility-Impact Optimum 25For F(x)=4 ln(1+x)-x 2 on x>-1, what x maximizes F and what is the maximum value?数学困难derivation未尝试免费1974Weighted Exposure Hedge Split 4The first hedge line loads twice as much on the required exposure as the second, but also has its own quadratic penalty. Minimize L(x,y) = 2x 2 + 3y 2 subject to 2x + 1y = 12.数学中等数值题未尝试免费1975Three-to-One Loading Tradeoff 5One book moves the target much faster per unit notional, so the optimizer must trade off load efficiency and cost. Minimize L(x,y) = 3x 2 + 1y 2 subject to 1x + 3y = 9.数学困难derivation未尝试面试订阅1984Why the Total-and-Spread Problem Has a Unique Solution 14Why does the problem min a x 2 + b y 2 + c z 2 subject to x+y+z=N and x-z=d have a unique optimizer when a,b,c>0?数学困难derivation未尝试面试订阅1990Minimum Risk Needed for a Chosen Alpha Level 20If a desk needs mu 1 x + mu 2 y = A with minimum quadratic risk a x 2 + b y 2, what minimum risk level is required?数学困难derivation未尝试面试订阅2034Conditional Jensen Lower Bound 14If phi is convex, what inequality holds between E[phi(X)|F] and phi(E[X|F]) almost surely?数学困难derivation未尝试面试订阅2039Convex Penalty for Mixed Schedules 19Two execution schedules have penalties phi(q 1) and phi(q 2) under a convex phi. What does Jensen say about a random 50-50 mix versus the penalty at the average size?数学困难derivation未尝试面试订阅2040Three-Scenario Square-Root Impact Gap 20Suppose V takes values 0, 3, and 8 with equal probability. Compute E[sqrt(1+V)] and sqrt(1+E[V]).数学困难数值题未尝试面试订阅2044Why Equality Holds Only Without Dispersion 24For a strictly convex phi, when can Jensen's inequality E[phi(X)] >= phi(E[X]) hold with equality?数学中等derivation未尝试免费2398Bias Budget Implied by a Variance ReductionA regularization change reduces a model's variance term from 0.30 to 0.11 while leaving irreducible noise unchanged. How much extra bias squared could you add before the total MSE stops improving?机器学习中等derivation未尝试免费2399Optimal Weight on a Noisy Unbiased ModelModel A is unbiased with variance 9. Model B has variance 1.44 and fixed bias 0.6. If you blend them as P w = wA + (1-w)B and treat their errors as independent, what weight w minimizes MSE?机器学习困难derivation未尝试面试订阅