Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 30K / a / Solution

Suppose $x\mid\mathcal F\sim N(\widehat x,V)$ and the new observation is $y=x+\eta$, where $\eta\sim N(0,R)$ is independent. Then
$$
\mathbb E[x\mid\mathcal F,y]
=\widehat x+\frac{V}{V+R}(y-\widehat x),
\qquad
\operatorname{Var}(x\mid\mathcal F,y)
=\frac{VR}{V+R}.
$$
The coefficient $V/(V+R)$ is the scalar Kalman gain.


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