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4458More Signals, Same DataIf you keep adding candidate signals without increasing data length, what often happens to the reliability of estimated combination weights?机器学习中等essay未尝试面试订阅4459Score Scale DriftIf one signal's score scale drifts over time while another remains stable, what usually happens to a fixed raw-score blend?机器学习中等essay未尝试面试订阅4460Slow Signal WeightIf transaction costs rise materially, what usually happens to the appeal of slower-moving signals in the blend?机器学习中等essay未尝试面试订阅4461Before CombiningBefore combining several signals, what should you check first besides each signal's standalone Sharpe?机器学习中等essay未尝试面试订阅4462Before Optimizing WeightsWhat should you inspect first before trusting an optimizer's exact signal weights?机器学习中等essay未尝试面试订阅4463Before Using Raw ScoresWhat is the first comparability question before blending raw signal scores?机器学习中等essay未尝试面试订阅4464Before Adding Meta-ModelingBefore fitting a meta-model on top of several signals, what is the first data question you should ask?机器学习中等essay未尝试面试订阅4465Before Declaring DiversificationWhat should you check first before saying that adding five more signals makes the ensemble diversified?机器学习中等essay未尝试面试订阅5858Integrating Factor with Variable CoefficientSolve the first-order linear ODE x'(t)+\dfrac x(t) t =t for t>0 with x(1)=2, and evaluate x(2).数学中等derivation未尝试面试订阅5859Separable Nonlinear ODE and Blow-UpSolve the separable ODE y'(x)=x\,y(x) 2 with y(0)=1. At what value of x>0 does the solution blow up?数学中等derivation未尝试面试订阅5860Overdamped Second-Order ODE with Distinct Real RootsSolve y''-5y'+6y=0 with y(0)=1 and y'(0)=0.数学中等derivation未尝试面试订阅5891Who Owns the Class PriorTwo teams ship classifiers trained on a balanced 50/50 dataset, but the live population is 90% class 0. Team A used Gaussian discriminant analysis; Team B used logistic regression. Which model explicitly contains an estimate of the class prior P(y), and explain why that distinction makes one team's fix to the prevalence mismatch cleaner than the other's.机器学习中等essay未尝试面试订阅5892Posterior from a Generative Gaussian ModelA generative classifier models one feature as Gaussian within each class with equal variance: x|Y=0 ~ N(0,1), x|Y=1 ~ N(2,1), and class prior P(Y=1)=0.5. Using Bayes' rule to convert this generative description into the discriminative posterior, compute P(Y=1|x=1.5).机器学习中等数值题未尝试面试订阅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5901Expected Rounds to Double a Kelly BankrollA gambler bets the Kelly fraction on an even-money coin with win probability p=0.6 every round, so log-wealth is a random walk with positive drift. Let G be the per-round expected log-growth (the maximal Kelly growth rate). Using an optional-stopping argument on a suitable martingale, estimate the expected number of rounds until wealth first doubles. You may ignore overshoot past the doubling level.概率困难数值题未尝试面试订阅5902Kelly Sizing with an Unknown Win ProbabilityA coin's win probability is unknown, with prior \sim Beta (2,2). You observe 7 wins and 3 losses in calibration trials, then must place one even-money bet on the next flip, choosing a fraction f of wealth to maximize the expected log-wealth after that bet. What fraction should you bet, and why is the posterior mean (rather than, say, the posterior mode) the right quantity to plug into the Kelly formula?概率中等数值题未尝试面试订阅5906How Many Bets Until Loss Is UnlikelyA Kelly bettor on an even-money coin with p=0.6 stakes the optimal fraction f *=0.2 each round. The per-round log-return is +\ln 1.2 with probability 0.6 and \ln 0.8 with probability 0.4, with mean G\approx0.0201 and variance v\approx0.0395. Using Chebyshev's inequality, find a number of rounds n after which the probability of ending below the starting wealth is at most 5\%.概率困难数值题未尝试面试订阅5911How Long Can You Play Before the Edge Eats YouYou start with \2 and bet \1 per round on an even-money game you win with probability p=0.4. You play until you either reach \5 or go broke. What is the expected number of rounds you play before the game ends?概率困难数值题未尝试面试订阅