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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3076Fusing Two Noisy Dealer QuotesA latent scalar state has prior N(0,9). Two conditionally independent sensors observe y 1=2 with noise variance 4 and y 2=-1 with noise variance 5. Compute the posterior mean and posterior variance after both observations.统计中等derivation未尝试面试订阅3077Two-Sensor Latent Level EstimateA latent scalar state has prior N(5,16). Two conditionally independent sensors observe y 1=9 with noise variance 9 and y 2=3 with noise variance 4. Compute the posterior mean and posterior variance after both observations.统计中等derivation未尝试面试订阅3078Dual Feed State CombinationA latent scalar state has prior N(-2,25). Two conditionally independent sensors observe y 1=-1 with noise variance 1 and y 2=2 with noise variance 4. Compute the posterior mean and posterior variance after both observations.统计中等derivation未尝试面试订阅3081Two Missing Days Before a Print ArrivesA local-level model satisfies x t=x t-1 +w t with w t\sim N(0,1), and observations have noise variance 2. After the last filtered state N(3,4), there are 2 consecutive missing observations. Then you observe a new value y=6. Compute the variance just before the new observation and the updated mean/variance after processing it.统计中等derivation未尝试面试订阅3082One Missing Observation Then UpdateA local-level model satisfies x t=x t-1 +w t with w t\sim N(0,3), and observations have noise variance 5. After the last filtered state N(-1,9), there are 1 consecutive missing observations. Then you observe a new value y=2. Compute the variance just before the new observation and the updated mean/variance after processing it.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3096Tomorrow Variance After a Large ShockIn a GARCH(1,1) model with \omega= 1 10 , = 1 5 , and = 7 10 , suppose the current squared return is r t 2=4 and the current conditional variance is h t=2. Compute h t+1 .统计简单derivation未尝试面试订阅3099Volatility Update from a Moderate ReturnIn a GARCH(1,1) model with \omega=1, = 3 20 , and = 3 5 , suppose the current squared return is r t 2=4 and the current conditional variance is h t=5. Compute h t+1 .统计简单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3106Half-Life When Alpha Plus Beta Equals 0.8In a GARCH-style volatility recursion, a deviation from long-run variance decays approximately by the factor = + = 4 5 each step. What is the half-life of the deviation?统计中等derivation未尝试面试订阅3111Does This GARCH Have a Finite Long-Run Variance?For a GARCH(1,1) model with \omega= 1 5 , = 1 4 , and = 3 4 , decide whether the model has a finite unconditional variance. If it does, compute it.统计中等derivation未尝试面试订阅