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A fixed state, repeated readings

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A constant unknown $x$ has a prior with variance $6$ . You take $2$ independent Gaussian readings of $x$ , each with noise variance $6$ . What is the posterior variance of $x$ ? Give a number.

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Hint

Gaussian precisions add, and the state does not move, so there is no process noise.

Solution

Each source has precision $\frac{1}{6}$ , and there are $3$ sources, counting the prior.
The total precision is $\frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}$ .
The posterior variance is $\frac{1}{\frac{1}{2}} = 2$ .

Answer: $2$

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