 Scalar static-state Kalman update

Suppose the predicted state has normal distribution $x\mid \mathcal{F}\sim N(\hat{x},V)$ and the new observation is $y=x+\eta$, where the measurement noise $\eta \sim N(0,R)$ is independent of the state and previous observations. Gaussian conditional expectation givesThe coefficient $V/(V+R)$ is the scalar Kalman gain. The posterior mean is a weighted mean of the predicted state and the new observation, with the more precise quantity receiving more weight. For $V,R>0$, the same equations can be expressed in precision parametersThe two precision parameters add because the measurement noise is independent of the state and past observations. Completing the square in the product of the prior density and the likelihood function gives this form directly. This is a Gaussian conditional expectation for a fixed unknown state; a moving state additionally requires a Kalman prediction step.