Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example given here: Resultant probability distribution when taking the cosine of gaussian distributed variable
This probably isn't a good question for mathoverflow but is more suited for math.stackexchange (which is for less advanced topics)
That said, here is the answer. You just reexpress the cosine function as a sum of two complex exponentials, and find that your answer can be expressed using the Fourier transform of the Gaussian density function. That fourier transform is known: it's another Gaussian, and so you have the solution to your question
