Given $\Lambda\sim\mathtt{Pois}\left(\lambda\right)$ and a stochastic process $X\left(t\right)=\Lambda\cos 2\pi t.$ Seek trend and covariance function

Question

Problem. Given $\Lambda\sim\mathtt{Pois}\left ( \lambda \right )$ and the stochastic process ${\it X}\left ( t \right )= \Lambda\cos 2\pi t.$ Seek the trend function and the covariance function of ${\it X}\left ( t \right )\!.$

Based on an old topic, I know the way to deal with it except the verification of equivalence $\mathbb{E}\left [ {\it X}\left ( t \right ) \right ]= \mathbb{E}\left [ \Lambda \right ]\mathbb{E}\left [ \cos 2\pi t \right ]{\it ?}$ I'm also a bit confused about what would covariance function work in this strange (to me) stochastic process. How simple is and I was forced to be doubtful. So I need your help.

Answer 1

$\cos (2\pi t)$ is not random. So $EX(t)=\cos (2\pi t) E\Lambda$ and $Var (X(t))=(\cos (2\pi t))^{2} Var \Lambda$. Also, $Cov (X(t),X(s))=\cos (2\pi t) \cos (2\pi s) Var (\lambda)$