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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5912All-In or Split the Stake to Double UpYou have \1 and want to double it to \2 on a subfair even-money bet that wins with probability p=0.45 (and you quit forever once you either reach \2 or hit \0). Compare two plans: (A) bet the whole \1 in one shot; (B) bet \0.50 each round until you reach \2 or go broke. Which gives the higher probability of doubling, and what are the two probabilities?概率困难数值题未尝试面试订阅5913Minimizing Ruin in a Favorable GameYou have 2 chips and play a favorable even-money game you win with probability p=0.6, intending to play forever (no cash-out target) and stake whole chips. To minimize the chance of ever going broke you bet the smallest stake, 1 chip per round. What is the probability you are eventually ruined under this minimum-stake (timid) play?概率中等数值题未尝试免费5915Timid Versus Bold to QuadrupleStarting with \1 you want to reach \4 on an even-money game you win with probability p=0.4, quitting when you reach \4 or go broke. Compute the probability of reaching \4 under (A) timid play, betting \1 each round, and (B) bold play, staking \min( current ,\ 4- current ). Which strategy gives the higher chance of reaching the goal?概率困难数值题未尝试面试订阅5916Most You Would Pay for a Perfect TestA product launch pays +30 if the market is receptive and -12 if it is not; receptivity has prior probability 3 10 . You may instead shelve the product for 0. A consultant offers a perfectly accurate test that reveals the true market state before you decide. What is the most you should be willing to pay for this test?概率中等derivation未尝试免费5917Free Peek Before Calling the Bigger BoxTwo boxes each independently contain an amount drawn uniformly from \ 1,2,3,4\ . You must guess which box holds the strictly larger amount; a correct guess pays 1 and a tie or wrong guess pays 0. Before guessing you may take a free peek at the contents of one box (your choice of which). By how much does this peek increase your probability of a correct guess compared with guessing blind?概率简单derivation未尝试免费5919One Free Draw Before Betting on the Majority ColorAn urn is type-R with probability \frac35 (then it is 80\% red balls) or type-B with probability \frac25 (then it is 80\% blue balls). You will bet on the urn's majority color: a correct bet pays 1, a wrong bet pays 0. You may first draw one ball (with replacement) and observe its color for free. By how much does observing this single draw raise your expected payoff over betting with no draw?概率中等derivation未尝试免费5920Clairvoyance Across Three StatesThe state is High, Mid, or Low with probabilities \frac12,\frac13,\frac16. You pick action Long or Flat once. Long pays 12,\ -3,\ -9 in High, Mid, Low respectively; Flat pays 0 in every state. A clairvoyant will tell you the exact state before you choose. What is the difference between your expected payoff acting on the clairvoyant's report and your expected payoff using the single best action chosen in advance?概率中等derivation未尝试免费5922Three-Candidate Best-ChoiceThree candidates of distinct, unknown qualities arrive in uniformly random order. After each interview you learn only the candidate's rank relative to those seen so far, and you must immediately and irrevocably hire or reject. You want to maximize the probability of hiring the single best candidate. What is the optimal policy and the resulting probability of success?概率简单数值题未尝试免费5981Total Stake Until the First SixYou roll a fair die repeatedly until the first time a 6 appears; let N be the number of rolls (including the winning roll). On each roll you independently collect a payoff X i with E[X i]=1.5, where the X i are i.i.d. and independent of the roll values. Compute E\! [\sum i=1 N X i ].概率简单derivation未尝试免费5983Number of Trades to Cross a Profit TargetA strategy books i.i.d. positive profits X 1,X 2,\dots with E[X i]=2.5. Let N be the first time the running total S n=\sum i=1 n X i strictly exceeds 10; that is, N=\min\ n:S n>10\ . Assuming E[N]< , the expected overshoot is known to satisfy E[S N]=14. Use a Wald-style identity to compute E[N].概率中等derivation未尝试免费5985Expected Total Slippage With Negative DriftA market-making desk incurs i.i.d. per-trade adverse-selection costs X 1,X 2,\dots with E[X i]=-0.4 (a net loss per trade). The number of trades in a session, N, is independent of the costs and is Poisson with mean 15. Compute the expected cumulative cost E\! [\sum i=1 N X i ].概率简单derivation未尝试免费5986Expected Winnings Over a Random Number of BetsA gambler places bets until a random stopping rule halts play; the number of bets N is a stopping time for the i.i.d. bet outcomes with E[N]=8. Each bet has an i.i.d. net result X i with E[X i]=-0.05 (a 5\% house edge per unit staked, with unit stakes), and the decision to stop after bet n depends only on outcomes up to bet n. Compute the gambler's expected total winnings E\! [\sum i=1 N X i ], and state whether any stopping rule with E[N]=8 can make this positive.概率中等derivation未尝试免费