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561Tagged Label Survives Unseen 1A particular tag is one of 9 equally likely outcomes on each of 11 independent draws. What is the probability that this tag is never drawn?概率简单数值题未尝试免费566Time to Reach 3 Distinct Types 1Coupons arrive uniformly from 7 types. What is the expected number of draws needed to see 3 distinct types for the first time?概率简单数值题未尝试免费571All Labels Seen by Draw 5 1Draw 5 coupons independently and uniformly from 3 types. What is the probability that all 3 types have appeared at least once by time 5?概率简单数值题未尝试免费576Discrete Signal Stop Rule 1You may inspect up to 2 independent signals, each uniformly distributed on 1,...,7 . After seeing a signal, you may lock it in and stop. Rejecting a signal and continuing costs 1 point(s). If you reach the last draw, you must take it. What first-round acceptance threshold is optimal, and what is the expected net score?概率简单数值题未尝试免费596Continuation Value Calibration 1A trader may inspect up to 4 independent candidate fills with support [2, 5, 9] and probabilities ['1/3', '1/3', '1/3']. Rejecting a fill and continuing costs 1. What first observation becomes just good enough to accept, and what is the overall optimal expected net value?概率简单数值题未尝试免费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1968Negative Softplus Tilt 23The saturation penalty dominates until x turns sufficiently negative. The desk maximizes K(x) = 1 x - 3 ln(1+e x). What x is optimal?数学中等derivation未尝试免费1983Derive the Total-and-Spread Solution 13For positive a,b,c, derive the minimizer of a x 2 + b y 2 + c z 2 subject to x+y+z=N and x-z=d.数学中等derivation未尝试免费1988How the Optimizer Scales With the Risk Radius 18In max mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2, how does the optimizer change when R is doubled?数学中等derivation未尝试免费2028Log Carry Score Is Concave 8Let psi(x)=ln(1+x) on x>-1. Compare E[psi(X)] and psi(E[X]).数学中等derivation未尝试面试订阅2037Why Jensen Matters for Nonlinear Risk Transforms 17Why is it dangerous to plug an average state into a nonlinear convex risk transform and treat that as the average transformed risk?数学中等essay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