第 1 / 2 页
非代码面试题
显示 20 / 24 道匹配题目
答题状态：未尝试未正确已正确
ID题目领域难度题型进度权限
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5897Overbetting to Twice KellyUnder the small-edge approximation G(f)\approx f-\tfrac12 2 f 2 for the expected log-growth of a repeated bet, the growth-optimal fraction is f *= / 2. At what (nonzero) betting fraction does the expected log-growth fall back to zero, and what does this say about the symmetry of growth around f *?概率中等数值题未尝试面试订阅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5905Kelly with a Cash-Reserve FloorYou may stake on an even-money coin with win probability p=0.8, but a liquidity rule requires you to keep at least half your total wealth in untouched cash at all times, so the staked fraction satisfies f\le 0.5. Set up the constrained maximization of expected log-growth, use the KKT conditions to determine the optimal stake, and state whether the reserve constraint binds.概率困难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5907Kelly with a Proportional Trading CostOn an even-money coin with win probability p, each round you pay a proportional cost c on the amount staked, regardless of the outcome. So staking fraction f, a win multiplies wealth by 1+f(1-c) and a loss by 1-f(1+c). Derive the growth-optimal fraction f * in terms of p and c, evaluate it for p=0.6,\ c=0.05, and find the cost level at which the optimal stake drops to zero.概率困难数值题未尝试面试订阅5908Reaching the Target Against a House EdgeYou hold 3 chips and bet one chip per round on an even-money game that you win with probability p=0.4 (and lose with probability 0.6). You keep playing until you either reach 5 chips (cash out) or hit 0 (broke). What is the probability you reach 5 chips before going broke?概率中等数值题未尝试免费5909Bold Play to a Quadrupling GoalYou have \2 and want to reach \8 on an even-money game you win with probability p=0.4. You use bold play: each round you stake the most that keeps you from overshooting the goal, i.e. \min( current wealth ,\ 8- current wealth ). What is the probability bold play reaches the \8 goal?概率中等数值题未尝试免费5910The Martingale Doubling System on RouletteOn a roulette red bet you win (even money) with probability 18/38 and lose with probability 20/38. You run the doubling (martingale) system aiming to win \1: bet \1; if it loses, bet \2; if that loses, bet \4. You stop after a win or after three straight losses (your bankroll of \7 is then gone). Find (a) the probability the campaign ends in ruin and (b) the expected net profit of the campaign.概率中等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