15.1.16 Theory of random processes

Problem: The random function \( X(t) \) has the characteristics \( m_{X_{t}}=\cos 3 t+t^{4}-2 t+4, K_{X}\left(t_{1}, t_{2}\right)=4 \cdot e^{6\left(t_{1}^{2}+t_{2}^{2}\right)} \). Find the expected value, the correlation function and the dispersion of the random function \( Y(t)=\sin 3 t \cdot X^{\prime}(t)+5 t^{2} X(t)+t^{3}+1 \).