805. Find the correlation function of the random function $Z(t)=X(t)+X^{\prime}(t)$, given the correlation function $K_{x}$.
Solution. By Theorem 2 ($\S 2$), $K_{z}=K_{x}+K_{\dot{x}}+R_{x \dot{x}}+R_{\dot{x} x}$. Considering that the correlation function of the derivative (Theorem 2)

$$
K_{\dot{x}}=\frac{\partial^{2} K_{x}}{\partial t_{1} \partial t_{2}}
$$

and the mutual correlation functions (Theorem 3)

$$
R_{x \dot{x}}=\frac{\partial K_{x}}{\partial t_{2}}, \quad R_{\dot{x} x}=\frac{\partial K_{x}}{\partial t_{1}}
$$

we finally obtain the desired correlation function:

$$
K_{z}=K_{x}+\frac{\partial^{2} K_{x}}{\partial t_{1} \partial t_{2}}+\frac{\partial K_{x}}{\partial t_{2}}+\frac{\partial K_{x}}{\partial t_{1}}
$$