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Let ${U(t)}$ be a gaussian random process.

Let $\phi $ be a random variable such that $\phi \sim \text{Uniform}(0,2\pi)$.

$U(t)$ is independent of $\phi $.

Let $Y(t)$ be a random process that created by $ Y(t) = U\left(t\right)\cdot\cos\left(2\pi f_{c}t+\phi\right) $

Is $Y(t)$ a gaussian random process?

I know that to show that a random process is gaussian I need to show that for every linear combination with times $(t_1,t_2,\dots t_n) $ the linear combination of : $$Q=\sum_{i=1}^{n}\alpha_{i}U\left(t_{i}\right)\cos\left(2\pi f_{c}t_{i}+\phi\right) $$ is a gaussian random variable.

But in this case the $\phi $ is random so how do I know if it's a gaussian random variable or not?

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I found the solution to the problem.

In general, a gaussian random variable $X$ with mean $E[X] = 0$ and $Var(X)=\sigma^2$ must satisfy: $$E[X^4]=3\sigma^4 $$ In our case we can calculate for $$Q = U(0)\cos(2 \pi f_c \cdot 0 + \phi)=U(0)\cos(\phi)$$ Calculating $E[Q^4]$ does not satisfy the constraint.

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