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3161Trade Only If Posterior Regime Probability Exceeds 70%A latent profitable regime has prior probability 1 2 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HH`, should you act if the required posterior threshold is 7 10 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3162Allocate Capital Only Above a 60% PosteriorA latent profitable regime has prior probability 2 5 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HHT`, should you act if the required posterior threshold is 3 5 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3164Stop Trading Unless the Posterior Stays Above 55%A latent profitable regime has prior probability 3 5 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HTT`, should you act if the required posterior threshold is 11 20 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3165Turn On the Hedge Above 65% PosteriorA latent profitable regime has prior probability 1 4 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HHHT`, should you act if the required posterior threshold is 13 20 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3176Aggressive vs Defensive Quote After a SignalThere are two possible actions. `Aggressive` pays 10 in the good state and -8 in the bad state. `Defensive` pays 4 in the good state and -1 in the bad state. The good state has prior probability 2 5 . Before acting, you may see a binary signal that is correct with probability 4 5 . What is the value of observing the signal, and which action should you take after a good signal and after a bad signal?概率困难derivation未尝试面试订阅3177Allocate Between Fast and Safe BooksThere are two possible actions. `Aggressive` pays 9 in the good state and -6 in the bad state. `Defensive` pays 5 in the good state and 1 in the bad state. The good state has prior probability 1 2 . Before acting, you may see a binary signal that is correct with probability 3 4 . What is the value of observing the signal, and which action should you take after a good signal and after a bad signal?概率困难derivation未尝试面试订阅3186Perfect Information Before Choosing a Desk StrategyTwo actions are available before the state is revealed. Action A pays 10 in the good state and -4 in the bad state. Action B pays 4 in the good state and 3 in the bad state. The good state has prior probability 2 5 . What is the expected value of perfect information about the state before acting?概率中等derivation未尝试面试订阅3921American Put in One-Step Tree IA one-step binomial model gives next-step option payoffs 0 in the up state and 14 in the down state, with risk-neutral up probability 0.45 and interest rate 5.00% for the step. The option's immediate exercise value today is 7. What are the European and American values today?金融与交易中等derivation未尝试面试订阅3924American Put in One-Step Tree IIIIn a one-step binomial tree, the stock is 80 today and moves to 88 or 64 next period. The strike is 85 and the one-period risk-free rate is 0.1. What is the American put value today?金融与交易中等derivation未尝试面试订阅3925American Call in One-Step Tree with No DividendA one-step binomial model gives next-step option payoffs 1 in the up state and 9 in the down state, with risk-neutral up probability 0.5 and interest rate 0.00% for the step. The option's immediate exercise value today is 4.5. What are the European and American values today?金融与交易中等derivation未尝试面试订阅3936Why Early Exercise Is an Economic Decision, Not a FormulaWhy is early-exercise logic best understood as a tradeoff among dividends, carry, and time value rather than a memorized checklist?金融与交易中等essay未尝试面试订阅3937Why American Flexibility Often Looks Cheap Until It Suddenly MattersWhy can the extra flexibility of an American option look irrelevant most of the time and then matter a lot in a narrow set of states?金融与交易中等essay未尝试面试订阅3938Why Puts and Calls Behave AsymmetricallyWhat is the cleanest reason American puts and American calls do not share the same early-exercise logic on standard equities?金融与交易中等essay未尝试面试订阅3939How to Sanity-Check an Early-Exercise AnswerWhat is a fast sanity check after you answer an early-exercise question in an interview?金融与交易中等essay未尝试面试订阅3940Why One-Step Trees Still MatterWhy are one-step binomial American-option questions still useful even for candidates who know much more advanced pricing models?金融与交易中等essay未尝试面试订阅5066Infer Self-Transition Probability From a Bellman Value 1Under a fixed policy, state s yields immediate reward 1 each step. With probability p it returns to s next step; otherwise the episode ends. If the discount factor is 0.9 and the state value is reported as V(s)=2.5, what p is implied?机器学习简单数值题未尝试面试订阅5067Infer Self-Transition Probability From a Bellman Value 2Under a fixed policy, state s yields immediate reward 0.5 each step. With probability p it returns to s next step; otherwise the episode ends. If the discount factor is 0.95 and the state value is reported as V(s)=2, what p is implied?机器学习简单数值题未尝试面试订阅5068Infer Self-Transition Probability From a Bellman Value 3Under a fixed policy, state s yields immediate reward 2 each step. With probability p it returns to s next step; otherwise the episode ends. If the discount factor is 0.8 and the state value is reported as V(s)=4, what p is implied?机器学习简单数值题未尝试面试订阅5069Infer Self-Transition Probability From a Bellman Value 4Under a fixed policy, state s yields immediate reward 1.2 each step. With probability p it returns to s next step; otherwise the episode ends. If the discount factor is 0.85 and the state value is reported as V(s)=2.4, what p is implied?机器学习简单数值题未尝试面试订阅5076Choose the Greedy Backup Action 11In one state, action 1 gives immediate reward 0.6 and then moves to states of value 3 with probability 0.4 and 1 otherwise. Action 2 gives immediate reward 0.9 and then moves to states of value 0.2 with probability 0.1 and 2 otherwise. If gamma=0.9, which action is greedy and what backup value does it produce?机器学习中等数值题未尝试面试订阅