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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6030One-Step Prediction with a Persistence CoefficientA latent state evolves as x t=0.9\,x t-1 +w t with w t\sim N(0,2). At time t-1 the filtered state is N(4,3). Compute the one-step-ahead predicted mean and predicted variance of x t (before any observation at time t).统计简单derivation未尝试免费6031Kalman Gain AloneIn a scalar measurement update y=x+\varepsilon with \varepsilon\sim N(0,4), the prior (predicted) state variance is P -=12. What fraction of the innovation is incorporated into the updated estimate, i.e. compute the Kalman gain K.统计简单数值题未尝试免费6032How Much Does One Print Shrink Uncertainty?A predicted state has variance P -=10. A single observation arrives with noise variance R=6 in the model y=x+\varepsilon. By how much does the posterior (updated) variance fall below P -? Give the updated variance P +.统计简单数值题未尝试免费6033Innovation Variance and a Standardized SurpriseIn the model y t=x t+v t with v t\sim N(0,3), the predicted state at time t is N(5,7). You then observe y t=11. Compute the innovation (one-step forecast error) variance S, and the standardized innovation (y t-m -)/ S .统计中等derivation未尝试面试订阅6034Where Does the Estimate Land After the Print?The predicted state is N(8,6) and you observe y=14 with measurement noise variance R=2 in y=x+\varepsilon. Compute only the updated (posterior) mean of the state.统计简单数值题未尝试免费