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](https://mathoverflow.net/questions/35260/resultant-probability-distribution-when-taking-the-cosine-of-gaussian-distribute)



expected value of cosine wirh Gaussian phase
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 


