We consider the one-step prediction problem for discrete-time linear systems in correlated plants and observation noises, and non-Gaussian initial conditions. Explicit representations are obtained for the MMSE and LMMSE (or Kalman) estimates of the state given past observations, as well as for the expected square of their difference. These formulae are obtained with the help of the Girsanov transformation for Gaussian white noise sequences, and display explicitly the dependency of the quantities of interest on the initial distribution. Applications of these results can be found in [5] and [6].