775. Find: a) the mathematical expectation; b) the correlation function; c) the variance of the random function $X(t)=U \cos 2 t$, where $U$ is a random variable, and $M(U)=5, D(U)=6$.
Solution. a) Let's find the required mathematical expectation (the non-random factor $\cos 2 t$ is taken out of the sign of the mathematical expectation):

$$
M[X(t)]=M[U \cos 2 t]=\cos 2 t M(U)=5 \cos 2 t,
$$

b) Let's find the centered function:

$$
\dot{X}(t)=X(t)-m_{x}(t)=U \cos 2 t-5 \cos 2 t=(U-5) \cos 2 t
$$

Let's find the required correlation function:

$$
\begin{gathered}
K_{x}\left(t_{1}, t_{2}\right)=M\left[\dot{X}\left(t_{1}\right) \dot{X}\left(t_{2}\right)\right]=M\left\{\left[(U-5) \cos 2 t_{1}\right]\left[(U-5) \cos 2 t_{2}\right]\right\}= \\
=\cos 2 t_{1} \cos 2 t_{2} M(U-5)^{2} .
\end{gathered}
$$

Considering that $M(U-5)^{2}=D(U)=6$, we finally have

$$
K_{x}\left(t_{1}, t_{2}\right)=6 \cos 2 t_{1} \cos 2 t_{2}
$$

c) Let's find the required variance by setting $t_{1}=t_{2}=t$:

$$
D_{x}(t)=K_{x}(t, t)=6 \cos ^{2} 2 t .
$$