Kalman filter: state estimate

Solution 1:

Each measurement is associated with a certain time index and this then also fixes the associated estimate of the state with respect to an index. For example $\hat{x}_{k|k} = E[x_{k}|z_{k}, \dots, z_{0}]$ is a different problem compared to $\hat{x}_{k+1|k} = E[x_{k+1}|z_{k}, \dots, z_{0}]$. Namely, in that case the measurement $z_{k+1}$ is not used, so during that time step one can only predict. For example if $w_k\sim\mathcal{N}(0,W)$ then the expected value of the prediction would yield $\hat{x}_{k+1|k} = F\,\hat{x}_{k|k}$.

Also note that the state itself is a stochastic process, which does not have to be completely "random" since it can be modeled by $x_{k} = F\,x_{k-1} + w_{k}$. Only if $F=0$ would the state inherit all the stochastic properties from $w_k$.