$8\mathrm{.}4-2$ Show that $R_{xy}(),$ the cross$-$correlation function of the input process $x(t)$ and the output process $y(t)$ in Fig. $8\mathrm{.}12,$ is

$R_{xy}()=h()**R_x(),$ and $,S_{xy}(f)=H(f)S_x(f)$

Hence, show that for the thermal noise $n(t)$ and the output $v_o(t)$ in Fig. $8\mathrm{.}13($Example $8\mathrm{.}9),$

$S_{n{v}_o}(f)=\frac{2kTR}{1+j2fRC},$ and $,R_{n{v}_o}()=\frac{2kT}{C}{e}^{-\frac{}{R}C}u()$