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245Maximum Entropy Property of the Normal DistributionThe differential entropy of a continuous random variable X with PDF f is h(X) = -\int - f(x) \ln f(x)\, dx. (a) Among all continuous distributions on R with mean and variance 2, use the method of Lagrange multipliers to show that the PDF maximizing h(X) satisfies \ln f(x) = -1 + \lambda 0 + \lambda 1 x + \lambda 2 x 2 for some constants \lambda 0, \lambda 1, \lambda 2. (b) Determine \lambda 0, \lambda 1, \lambda 2 by enforcing the three constraints (\int f = 1, \int xf = , \int x 2 f = 2 + 2) and show that f is the N( , 2) PDF. (c) Compute h(X) for X \sim N( , 2).概率困难derivation未尝试免费1790Ridge Effective Degrees of Freedom 5A standardized ridge model has singular-value squares d j 2 = [12.25, 4, 0.25] and penalty lambda = 0.25. What is the effective degrees of freedom tr(S lambda) = sum d j 2/(d j 2+lambda)?统计困难derivation未尝试面试订阅1971Cheapest Two-Book Hedge 1Two hedge books carry different quadratic slippage penalties but must deliver one joint exposure target. Minimize L(x,y) = 1x 2 + 4y 2 subject to 1x + 2y = 10.数学简单数值题未尝试免费1972Derive the Two-Book Hedge Formula 2Derive the minimizer of a x 2 + b y 2 subject to u x + v y = c for positive a,b.数学简单derivation未尝试免费1973How the Cheap Book Gets More Size 3In the problem min a x 2 + b y 2 subject to x+y=c, which book gets the larger allocation when a<b, and why?数学中等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1976Three-Book Budget Allocation 6Three sleeves have different quadratic slippage penalties but must add up to a fixed total size. Minimize Q(x,y,z) = 1x 2 + 2y 2 + 4z 2 subject to x+y+z = 28.数学简单数值题未尝试免费1977Derive the Three-Book Total-Size Allocation 7Derive the minimizer of a x 2 + b y 2 + c z 2 subject to x+y+z=N for positive a,b,c.数学中等derivation未尝试免费1978Why the Three-Book Solution Is Inverse-Coefficient Weighted 8In min a x 2 + b y 2 + c z 2 subject to x+y+z=N, why do larger quadratic coefficients receive smaller allocations?数学中等derivation未尝试免费1979Minimum Objective Value Under a Total-Size Constraint 9For positive a,b,c, what is the minimum value of a x 2 + b y 2 + c z 2 subject to x+y+z=N?数学中等derivation未尝试免费1980Minimum Risk Needed for a Target Alpha 10A desk wants to minimize a x 2 + b y 2 subject to mu 1 x + mu 2 y = A. What is the minimum achievable value?数学困难derivation未尝试面试订阅1981Three-Book Split With a Fixed Spread 11The first and third books must keep a pre-agreed spread while total size stays fixed. Minimize 1x 2 + 1y 2 + 1z 2 subject to x+y+z=9 and x-z=1.数学简单数值题未尝试免费1982Asymmetric Spread-Constrained Allocation 12The left book is more expensive per unit, so the optimizer cannot simply split the spread evenly. Minimize 2x 2 + 1y 2 + 1z 2 subject to x+y+z=10 and x-z=2.数学简单数值题未尝试免费1983Derive the Total-and-Spread Solution 13For positive a,b,c, derive the minimizer of a x 2 + b y 2 + c z 2 subject to x+y+z=N and x-z=d.数学中等derivation未尝试免费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1986Alpha Maximization on a Unit-Risk Ellipse 16Two sleeves carry different expected edges but must lie on one fixed quadratic risk budget. Maximize 3x + 4y subject to 1x 2 + 1y 2 = 25.数学简单数值题未尝试免费1987Derive the Alpha-Maximizing Point Under a Quadratic Risk Budget 17Derive the maximizer of mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2 for positive a,b.数学中等derivation未尝试免费1988How the Optimizer Scales With the Risk Radius 18In max mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2, how does the optimizer change when R is doubled?数学中等derivation未尝试免费1989Maximum Alpha Under a Quadratic Risk Budget 19What is the maximum value of mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2?数学困难derivation未尝试面试订阅