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4293Weight Decay Shrinkage 3A parameter has current value w = 2.0 and gradient g = 0.3. Using a decoupled weight-decay update w new = (1 - eta*lambda) w - eta*g with eta = 0.1 and lambda = 0.05, what is the updated weight after one step?机器学习简单数值题未尝试面试订阅4294Weight Decay Shrinkage 4A layer weight vector is w = (3, 4), so its norm is 5. You enforce max-norm regularization with cap c = 4 by rescaling only when the norm exceeds c. What vector is stored after clipping?机器学习简单数值题未尝试面试订阅4295Weight Decay Shrinkage 5An optimizer uses the proximal L1 shrinkage step sign(w)*max(|w| - tau, 0). If the pre-step weight is w = 0.7 and tau = 0.2, what weight remains after shrinkage?机器学习简单数值题未尝试面试订阅4446Equal Variance Contribution Weight 6Two independent signal sleeves have standard deviations 2 and 1. In a composite C = w S1 + (1-w) S2, what weight w makes the two sleeves contribute equally to the total variance?机器学习中等数值题未尝试面试订阅4447Implied Covariance From Chosen Blend 7A desk uses C = 0.7 S1 + 0.3 S2. The standard deviations of S1 and S2 are 1.0 and 1.5, and the standard deviation of C is 0.95. What covariance between S1 and S2 is implied?机器学习中等数值题未尝试面试订阅4448Correlation Shock Benefit 8An equal-weight composite combines two standardized signals. If their correlation drops from 0.6 to 0.2, by how much does the composite standard deviation fall?机器学习中等数值题未尝试面试订阅4449Target Alpha Weight 9A fast signal has expected alpha 9 bps and a slow signal has expected alpha 3 bps. In a composite C = w fast + (1-w) slow, what weight on the fast signal produces expected alpha 6.6 bps?机器学习中等数值题未尝试面试订阅4450MSE Gain From A Diversifying Forecast 10Forecast error variance is 4 for model A, 9 for model B, and their error covariance is 1. You blend them equally. By how much does the blended forecast's MSE improve relative to using model A alone?机器学习中等数值题未尝试面试订阅5289Why Long-Only Can Reshape The Entire SolutionAn unconstrained optimizer wants a large short in one equity sleeve to hedge two crowded longs, but mandate rules require long-only weights. Explain why the long-only solution can look qualitatively different rather than just a clipped version of the unconstrained one.金融与交易困难essay未尝试面试订阅5790Optimal Width Under Linear Fill DecayPer-side fill probability falls linearly with half-spread: p(h) = 0.6 - 4*h for h in [0, 0.15]. Net edge per fill is (h - loss) with loss = 0.01. Expected single-side PnL per round is p(h)*(h - loss). What half-spread h maximizes it?金融与交易中等数值题未尝试面试订阅5823Optimal Skew After A FillAfter a round-1 buy fill leaves you long 1 unit, you choose a round-2 quote skew s (in cents) that shifts both quotes down to encourage a sell. Selling probability is 0.3 + 0.1*s and expected edge per sell is 0.05 - 0.01*s, for s between 0 and 5. Round-2 expected edge is (selling probability)*(edge). Which integer s in 0,1,2,3,4,5 maximizes round-2 expected edge?金融与交易困难数值题未尝试面试订阅5893Deriving the Even-Money Kelly FractionYou repeatedly bet a fraction f of your current wealth on an even-money wager that wins with probability p>\tfrac12 (you gain the staked amount on a win, lose it on a loss). By maximizing the expected logarithm of your wealth multiplier over one round, derive the growth-optimal fraction f *.概率简单derivation未尝试免费5894Kelly Fraction at General Net OddsA favorable bet pays net odds b to 1: staking an amount, you gain b times the stake with probability p and lose the stake with probability 1-p. Betting a fraction f of wealth each round, derive the growth-optimal fraction f * in terms of b and p.概率简单derivation未尝试免费5895Maximum Growth Rate of a Kelly BettorAn even-money coin wins with probability p=0.6. You bet the growth-optimal (Kelly) fraction every round. Compute the resulting maximum expected log-growth rate per round, and express it in closed form in terms of p.概率中等数值题未尝试免费5896Why Half-Kelly Keeps Three-Quarters of the GrowthFor a small-edge repeated bet the expected log-growth is well approximated by the quadratic G(f)\approx f-\tfrac12 2 f 2, where and 2 are the per-round mean and variance of the bet's return. Using this approximation, find the optimal fraction f * and show what fraction of the maximal growth G(f *) is retained by betting half-Kelly, f=f */2.概率中等derivation未尝试面试订阅5898Continuous Kelly for Normal ReturnsEach round you allocate a fraction f of wealth to a position whose one-period return R is approximately normal with small mean >0 and variance 2 (with 2\ll 2), so post-round wealth is multiplied by 1+fR. Using a second-order expansion of the log, derive the growth-optimal fraction f *.概率中等derivation未尝试面试订阅5899Betting Kelly on the Wrong ProbabilityAn even-money coin truly wins with probability p=0.55, but you overestimate it as p=0.65 and bet the Kelly fraction implied by your estimate. What is your actual long-run expected log-growth rate per round? Compare it to the growth you would have earned betting the correct Kelly fraction, and state what the sign of your actual growth implies.概率困难数值题未尝试面试订阅5900Higher Expected Return, Lower Compounded GrowthAn even-money coin wins with probability 0.6. Trader A always stakes the fraction f A=0.10 of wealth; Trader B always stakes f B=0.40. (i) Whose stake has the higher one-round expected (arithmetic) profit? (ii) Whose wealth compounds faster over many rounds? Explain the apparent conflict.概率中等数值题未尝试免费5903Capping the Single-Bet DrawdownYou bet a fraction f of wealth on an even-money coin with win probability p=0.65, but a risk rule forbids any single losing bet from cutting your wealth by more than 20\%. What fraction should you bet, and for which win probabilities p does this drawdown rule actually constrain you below the Kelly fraction?概率简单数值题未尝试免费5904Kelly Exceeds Full InvestmentA favorable bet has limited downside: staking a fraction f of wealth, you gain the full amount f with probability p=0.7 but lose only half the stake, 0.5f, with probability 0.3. (a) Find the growth-optimal fraction f *. (b) If you cannot borrow (so f\le 1, i.e. you can stake at most all your wealth), what fraction do you actually bet?概率简单数值题未尝试免费