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3133Posterior Mean from Three Noisy MeasurementsSuppose \sim N(-1,16) and conditional on , you observe n=3 independent measurements with known variance 2=1 and sample mean x=2. Compute the posterior mean and posterior variance of .统计中等derivation未尝试面试订阅3136Posterior Predictive Category ProbabilitiesA categorical distribution over three states has prior Dirichlet (2,3,5). After observing counts [4, 1, 5], what is the posterior predictive probability that the next observation is in category 1?统计中等derivation未尝试面试订阅3141Posterior After HHTTA Bernoulli success probability has prior Beta (1,1). Data arrive sequentially with outcome string `HHTT` where each non-`T` symbol is a success and each `T` is a failure. What is the posterior after processing the full sequence, and what is the posterior predictive probability of success on the next trial?统计中等derivation未尝试面试订阅3146Daily Count Posterior After Three DaysA Poisson rate has prior Gamma (2,1) in shape-rate form. You observe sequential unit-time counts [3, 4, 2]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3147Posterior Rate After Four Trading SessionsA Poisson rate has prior Gamma (1,2) in shape-rate form. You observe sequential unit-time counts [1, 0, 2, 3]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3148Online Call-Rate UpdateA Poisson rate has prior Gamma (4,1) in shape-rate form. You observe sequential unit-time counts [5, 6]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3150Posterior Mean After Five Intraday BucketsA Poisson rate has prior Gamma (2,2) in shape-rate form. You observe sequential unit-time counts [0, 1, 2, 1, 3]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3151Posterior Mean After Two Streaming MeasurementsA latent mean has prior N(0,4). Observations arrive one by one with known noise variance 2=1 and realized values [2, 4]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3152Posterior Mean After Three Sequential Forecast ErrorsA latent mean has prior N(5,9). Observations arrive one by one with known noise variance 2=4 and realized values [6, 5, 3]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3153Online Update of a Latent DriftA latent mean has prior N(-1,16). Observations arrive one by one with known noise variance 2=1 and realized values [0, 2]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3155Posterior Mean After Four Noisy SignalsA latent mean has prior N(10,25). Observations arrive one by one with known noise variance 2=4 and realized values [12, 11, 8, 9]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3156Posterior Odds After Three Independent SignalsA binary hypothesis has prior probability 1 2 . Independent signals arrive sequentially with Bayes factors [2, Fraction(1, 2), 3] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?统计中等derivation未尝试面试订阅3157Posterior Odds Under Mixed EvidenceA binary hypothesis has prior probability 1 3 . Independent signals arrive sequentially with Bayes factors [4, Fraction(1, 5)] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?统计中等derivation未尝试面试订阅3158Posterior Probability After Bayes Factors 5 and 2A binary hypothesis has prior probability 2 5 . Independent signals arrive sequentially with Bayes factors [5, 2] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?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