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A hidden level follows $x_{t} = x_{t - 1} + w_{t}$ with $Var (w_{t}) = 3$ , and each reading is $y_{t} = x_{t} + v_{t}$ with $Var (v_{t}) = 24$ . The current estimate is $100$ with variance $P = 5$ . The next reading is $112$ . What is the updated estimate? Give a number.

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Hint

Predict the variance first, then compute the gain.

Solution

The predicted variance is $5 + 3 = 8$ .
The gain is $\frac{8}{8 + 24} = \frac{1}{4}$ .
The update is $100 + \frac{1}{4} \times (112 - 100) = 103$ .

Answer: $103$

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