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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5955Buy Random or Buy the Missing OneYou need all 5 types and currently hold 4 distinct types (exactly one type missing). Each round you may either (a) buy a random coupon for 1 (uniform over all 5 types), or (b) directly buy your missing type from a reseller for 5. Acting optimally to minimize expected total future cost, what is your minimum expected cost to complete the set?概率中等数值题未尝试免费