= Solution
Conditionally on past observations, adding the <independent> state increment gives the prediction
$$
S_t\mid\mathcal F_{t-1}\sim N(\widehat S_{t-1},R_t),
\qquad R_t=P_{t-1}+W.
$$
The new observation then has <conditional distribution> $N(\widehat S_{t-1},R_t+V)$, and its conditional <covariance> with $S_t$ is $R_t$. Equivalently, multiplying the state prediction <probability density function> by the observation <likelihood> and completing the square gives
$$
\frac1{P_t}=\frac1{R_t}+\frac1V,
\qquad
\frac{\widehat S_t}{P_t}=\frac{\widehat S_{t-1}}{R_t}+\frac{X_t}V.
$$
For positive <variances>, rearrangement yields the <scalar Gaussian Kalman recursion>
$$
\boxed{K_t=\frac{P_{t-1}+W}{P_{t-1}+W+V},\quad
\widehat S_t=\widehat S_{t-1}+K_t(X_t-\widehat S_{t-1}),\quad
P_t=\frac{V(P_{t-1}+W)}{P_{t-1}+W+V}.}
$$
Here the innovation is the observation minus its predicted <mean>. The displayed <covariance> formula also handles zero <variances> whenever its denominator is nonzero; a completely deterministic prediction and observation require no probabilistic update. <Independence> of the new noises from the past and the initial state is the usual state-space assumption needed for this conditional calculation.
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