15.1.17 Theory of random processes

Problem: Find the expected value, correlation function and the dispersion of the random process \( Z(t)=t^{2} \cdot X(t)++Y^{\prime}(t) \), if the random processes \( X(t) \) and \( Y(t) \) are uncorrelated, with the characteristics: \[ \begin{array}{l} m_{X}(t)=\sin 7 t, \quad m_{Y}(t)=4 t^{3}-5 t^{2}, \\ K_{Y}\left(t_{1}, t_{2}\right)=t_{1}^{3} \cdot t_{2}^{3} \cdot e^{2\left(t_{1}^{2}+t_{2}{ }^{2}\right)} . \end{array} \]