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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3166Signal Before a Binary TradeA trade pays +8 in a favorable state and -5 in an unfavorable state. The favorable state has prior probability 2 5 . Before trading, you may buy a signal for cost 1 2 ; it is correct with probability 4 5 . If you see the signal, you may either trade or abstain after observing it. What is the value of the signal, and should you buy it at that cost?概率中等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3191Total PnL Until a Geometric Number of FillsLet X 1,X 2,\dots be i.i.d. increments with E[X i]=3 and Var (X i)=5. Let N be independent of the increments and distributed as Geometric( 1 4 ) on 1,2,\dots . For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3192Aggregate Slippage Over a Poisson Number of OrdersLet X 1,X 2,\dots be i.i.d. increments with E[X i]=2 and Var (X i)=3. Let N be independent of the increments and distributed as Poisson(4). For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3193Total Cost Over a Negative-Binomial HorizonLet X 1,X 2,\dots be i.i.d. increments with E[X i]=4 and Var (X i)=6. Let N be independent of the increments and distributed as NegativeBinomial(r=3, p= 2 5 ). For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3201Expected Trials to Reach 5 SuccessesIndependent Bernoulli trials succeed with probability 2 5 . Let T be the first time the cumulative number of successes reaches 5. Use Wald-style reasoning to compute E[T].概率中等derivation未尝试面试订阅3206Variance of Trials to Reach 5 SuccessesIndependent Bernoulli trials succeed with probability 2 5 . Let T be the first time the cumulative number of successes reaches 5. Use Wald-style second-moment reasoning to compute Var (T).概率困难derivation未尝试面试订阅3212Second Moment of Centered Sum at a Poisson HorizonLet X 1,X 2,\dots be i.i.d. with mean and variance 3. Let N be independent of the increments and distributed as Poisson(4). Show that for the centered stopped sum M N=\sum i=1 N (X i- ), one has E[M N 2] equal to what value?概率中等derivation未尝试面试订阅3214Centered Slippage Variance Under Random StoppingLet X 1,X 2,\dots be i.i.d. with mean and variance 4. Let N be independent of the increments and distributed as Geometric( 1 3 ). Show that for the centered stopped sum M N=\sum i=1 N (X i- ), one has E[M N 2] equal to what value?概率中等derivation未尝试面试订阅3216Can a 95% Credible Interval Be Read as a 95% Probability Statement?A PM sees a 95% Bayesian credible interval for a strategy's daily edge and says, "So there is a 95% probability the true edge lies in this interval." Is that interpretation correct? Contrast it with the frequentist interpretation of a 95% confidence interval.统计中等essay未尝试面试订阅3217Why Optional Stopping Breaks a Fixed-Horizon p-ValueAn experiment is designed for a fixed horizon, but the desk checks the p-value every day and stops once it drops below 0.05. Why does that invalidate the nominal 5% Type I error guarantee? Why is the Bayesian posterior update answering a different question?统计中等essay未尝试面试订阅3218Small Sample, One Success: Which Lens Shrinks More Naturally?A new quoting rule is tried 5 times and succeeds once. Why does a Bayesian analysis with a skeptical prior naturally shrink the estimated success probability toward a baseline, while a plain frequentist point estimate does not do that unless you add some regularization device on top?统计中等essay未尝试面试订阅3219Posterior Probability of Positive Effect Versus p-ValueIf a PM wants to know, "What is the probability the treatment effect is positive?", why is a Bayesian posterior probability directly aligned with that question while a p-value is not?统计中等essay未尝试面试订阅3220Why a 'Weakly Informative' Prior Still MattersA researcher says, "I used a weakly informative prior, so the Bayesian answer is basically prior-free." Why is that statement too strong, especially in small or noisy samples?统计中等essay未尝试面试订阅