15.1.21 Theory of random processes

Problem: The random function \( X(t) \) has the characteristics \( m_{X}(t)=t^{5}+t \sin 2 t+2, K_{X}\left(t_{1}, t_{2}\right)=4 e^{-2\left(t_{1}-t_{2}\right)^{2}} \). Find the expected value, correlation function and the dispersion of the random process \( Y(t)=\cos t \cdot X(t)+3 t \cdot X^{\prime}(t)+4 t \).