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606Target-Hitting Stake Choice 6You start with wealth 5. In each of at most 3 rounds, you may bet any integer stake between 0 and your current wealth on an even-money coin that wins with probability 3/5. If you win, your wealth increases by the stake; if you lose, it decreases by the stake. What first-round stake maximizes the probability of finishing with wealth at least 9 after 3 rounds, and what is that maximal probability?概率简单数值题未尝试免费1584Minimum Unit-Norm Penalty 4What is the minimum value of 6x 2 + 8xy + 10y 2 over all vectors satisfying x 2 + y 2 = 1?数学中等derivation未尝试免费1594Residual Coefficient After Completing the Square 4Rewrite 8x 2 + 12xy + 7y 2 in the form 8(x + 3y/4) 2 + cy 2. What is c?数学困难数值题未尝试面试订阅1596Linear Term Implied by a Target Optimizer 1For f(x) = 0.5 x T A x - b T x with A = [[4,1],[1,3]], suppose the optimizer is x* = (1,2). What vector b makes that true?数学简单derivation未尝试免费1598Second Coordinate of a Coupled Quadratic Optimizer 3For f(x) = 0.5 x T A x - b T x with A = [[6,-2],[-2,4]] and b = (8,0), what is the optimizer's second coordinate?数学中等derivation未尝试免费1599Optimizer of a Coupled Positive-Definite Penalty 4Minimize f(x,y) = 0.5(3x 2 + 4xy + 5y 2) - (6x + 8y). What is the optimizer?数学中等数值题未尝试免费1600Coupling Needed for Equal Optimizer Coordinates 5For f(x) = 0.5 x T A(t) x - b T x with A(t) = [[5,t],[t,4]] and b = (6,2), for what t does the optimizer satisfy x1 = x2?数学困难derivation未尝试面试订阅1776Lasso Threshold Calibration 1A standardized lasso fit has score vector (4.1, 2.3, 1.7). What is the smallest lambda that makes every coefficient exactly zero?统计中等derivation未尝试免费1779Lasso Threshold Calibration 4In an orthonormal lasso update, a coordinate has score z = -3.2 and penalty lambda = 0.7. What coefficient results after soft-thresholding?统计中等derivation未尝试免费1793Elastic Net GroupingTwo features are almost duplicates but both are economically meaningful. Why does Elastic Net often behave better than pure Lasso here?统计中等derivation未尝试免费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1949Derive the Optimizer for Log Utility Minus Quadratic Cost 4For parameters a>0 and b>0, derive the unique maximizer of F(x)=a ln(1+x)-b x 2 on x>-1.数学中等derivation未尝试免费1950Quarter-Size Utility Balance 5A PM uses a very steep quadratic penalty, so the optimal tilt should stay small even with positive edge. The desk chooses a scalar exposure x > -1 to maximize F(x) = 5 ln(1+x) - 8 x 2. What exposure x maximizes F?数学困难derivation未尝试免费1951Bid-Ask Skew Balance 6On one quote axis, the maker gets more value from aggressive bids than from aggressive offers. A market maker chooses a skew x in (-1,1) to maximize G(x) = 5 ln(1+x) + 3 ln(1-x). What skew is optimal?数学简单数值题未尝试免费1952Derive the Optimizer for an Asymmetric Log Skew Objective 7For a>0 and b>0, derive the unique maximizer of G(x)=a ln(1+x)+b ln(1-x) on x in (-1,1).数学中等derivation未尝试免费1953Mild Positive Skew From Asymmetric Flow 8Order-flow asymmetry is present but not extreme, so the optimal skew should stay close to flat. A market maker chooses a skew x in (-1,1) to maximize G(x) = 7 ln(1+x) + 5 ln(1-x). What skew is optimal?数学中等derivation未尝试免费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