I am learning control engineering for some time and I work with a lot of transfer functions and frequency domain design. Reading from textbook, to me everything seems deterministic. Whenever I come across noise, which is either measurement noise or process noise, I typically believe that their effects cancel out in long term in case of Zero mean process. However, when I started reading stochastic process or kalman filters, I see a lot of significance given to statistical properties of signals like mean, variance, correlation, and other moments etc. Having spent so much time on deterministic world, now I cannot understand importance of stochastic world. For example, If someone where to show me a stochastic process modeled as a state space equation with process innovation and ask to predict future events, I would say the mean of the process can be determined by deterministic part of SS equation. The process noise can be ignored since we can't predict anything about it unless it is not zero mean. What does it matter about the variance, or correlation of thr signal? Similarly, if one were to show noisy signal, I would ask to average out n past signals points to remove noise, or if we know spectral information of noisr, I would ask to design appropriate rejection filter and pass signal to it. Again can't think of a reason to put statistical properties here.
I know I am missing something here given that soo much I read about stochastic process, but can't understand what.