After a short summary of scalar and vector random variables, the description of scalar and vector stochastic processes by probability distribution and density functions, expectations like correlation and covariance functions/matrices are considered. For stationary processes, further subjects are ergodicity, time averages, spectral density and correlation matrices.

The next chapter deals with some rule for matrices: Derivation to vectors and matrices, the pseudoinverse for solution or least-squares estimation of consistent or inconsistent linear equations, matrix inversion lemma.

A chapter on estimation theory deals with the methods of Bayes estimation (includig minimum variance, maximum a posteriori), maximum likelihood, and least-squares.

Based on these fundamentals, the equations of the time-discrete optimal filter (Kalman filter) for linear systems with normally distributed disturbances are derived (resp. optimal linear filter for arbitrary distribution). Numerical variants of the algorithm, as well as extensions (correlated system and measurement noise, coloured noise, continuous Kalman Bucy filter) are presented. For linear time-invariant systems, the relations between Kalman filter, Wiener filter, ans classic state observers are shown.

A short outlook deals with prediction, smoothing, and nonlinear filtering, followed by parameter estimation for linear system identification.

Finally, various application examples are presented.