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3061Expected Number of Arrivals in the First 20 Minutes Given Nine TotalA Poisson process is observed on [0,1] hour. Conditional on exactly 9 arrivals in the hour, what is the expected number of arrivals in the first 20 minutes?概率简单derivation未尝试面试订阅3062Expected Number of Arrivals in the Middle Half Given Eight TotalA Poisson process is observed on [0,1] hour. Conditional on exactly 8 arrivals in the hour, what is the expected number of arrivals in the middle 30 minutes?概率简单derivation未尝试面试订阅3063At Least One Arrival in the Final Quarter Given Four TotalA Poisson process is observed on [0,1] hour. Conditional on exactly 4 arrivals in the hour, what is the probability that at least one arrival falls in the final quarter-hour?概率中等derivation未尝试面试订阅3064Expected Idle Time Before the First Arrival Given Three TotalA Poisson process is observed on [0,1] hour. Conditional on exactly 3 arrivals in the hour, what is the expected idle time from 0 until the first arrival?概率中等derivation未尝试面试订阅3065Expected Idle Time After the Last Arrival Given Five Total in Two HoursA Poisson process is observed on [0,2] hours. Conditional on exactly 5 arrivals in the two-hour horizon, what is the expected idle time from the last arrival until hour 2?概率中等derivation未尝试面试订阅3066Signal Extraction from One Noisy PrintA latent scalar state has prior x\sim N(10,4). You observe y=13 through y=x+\varepsilon with \varepsilon\sim N(0,5). Compute the Kalman gain, posterior mean, and posterior variance.统计中等derivation未尝试面试订阅3068Latent Fair-Value UpdateA latent scalar state has prior x\sim N(-1,16). You observe y=3 through y=x+\varepsilon with \varepsilon\sim N(0,9). Compute the Kalman gain, posterior mean, and posterior variance.统计中等derivation未尝试面试订阅3071Local-Level Forecast Then UpdateSuppose x t=x t-1 +w t with w t\sim N(0,2), and y t=x t+v t with v t\sim N(0,3). At time t-1 the filtered state is N(7,4). You observe y t=9. Compute the predicted mean/variance and the updated mean/variance at time t.统计中等derivation未尝试面试订阅3072Random-Walk Value Filter StepSuppose x t=x t-1 +w t with w t\sim N(0,1), and y t=x t+v t with v t\sim N(0,4). At time t-1 the filtered state is N(-2,5). You observe y t=0. Compute the predicted mean/variance and the updated mean/variance at time t.统计中等derivation未尝试面试订阅3086Steady-State Gain with Q=1, R=2Consider the scalar local-level model in steady state: x t=x t-1 +w t, w t\sim N(0,1), and y t=x t+v t, v t\sim N(0,2). Compute the steady-state posterior variance C and the steady-state Kalman gain K.统计困难derivation未尝试面试订阅3091Long-Run Variance of a Quiet GARCH ProcessFor a GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 with \omega= 1 10 , = 1 5 , and = 3 5 , assume + <1. Compute the unconditional variance E[h t].统计中等derivation未尝试面试订阅3093Steady Variance from Daily GARCH ParametersFor a GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 with \omega=1, = 1 10 , and = 4 5 , assume + <1. Compute the unconditional variance E[h t].统计中等derivation未尝试面试订阅3101Two-Step Forecast from Today’s VarianceFor a GARCH(1,1) process with \omega= 1 10 , = 1 5 , = 3 5 , suppose you already know the one-step-ahead conditional variance h t+1 =2. Compute E t[h t+2 ] and E t[h t+3 ].统计中等derivation未尝试面试订阅3103Two-Day Ahead Variance MeanFor a GARCH(1,1) process with \omega=1, = 1 10 , = 4 5 , suppose you already know the one-step-ahead conditional variance h t+1 =5. Compute E t[h t+2 ] and E t[h t+3 ].统计中等derivation未尝试面试订阅3116Posterior Buy Probability After 7 Buys and 3 SellsA Bernoulli success probability p has prior Beta (2,3). After observing 7 successes and 3 failures, what is the posterior mean of p?统计简单derivation未尝试面试订阅3123Probability the Next Fill Is a SuccessA Bernoulli success probability has prior Beta (3,5). After observing 6 successes and 2 failures, compute the requested posterior predictive probability for the next 1 trial(s).统计中等derivation未尝试面试订阅3126Posterior Mean Arrival Rate After 12 Events in Two HoursA Poisson rate has prior Gamma (3,1) in shape-rate form. After observing 12 events over 2 hour(s), what is the posterior mean of ?统计中等derivation未尝试面试订阅3128Next-Hour Predictive Mean from Gamma-PoissonA Poisson rate has prior Gamma (4,1) in shape-rate form. After observing 5 events over 1 hour(s), what is the posterior predictive mean number of events in the next 1 hour(s)?统计中等derivation未尝试面试订阅3130Predictive Mean Trade Count in the Next Half HourA Poisson rate has prior Gamma (5,2) in shape-rate form. After observing 10 events over 4 hour(s), what is the posterior predictive mean number of events in the next 1 2 hour(s)?统计中等derivation未尝试面试订阅3131Posterior Mean of a Latent Fair ValueSuppose \sim N(0,4) and conditional on , you observe n=1 independent measurements with known variance 2=2 and sample mean x=3. Compute the posterior mean and posterior variance of .统计中等derivation未尝试面试订阅