I've got a quick question. I've been set this question:
To find the mean, cov, var I guess you consider them as two separate RP's so:
$$E(Y(t)) = \int \cos(\omega t+\psi)p_1(\psi)\,d\psi + \int \sin(\omega t+\gamma)p_1(\gamma)\,d\gamma$$
However I don't have any limits to work with so I'm unsure what I should be doing. Am I supposed to find limits from the conditions - if so, how? Thanks.
$\newcommand{\E}{\operatorname{E}}$I will assume that by $\overline{\cos\gamma}$ you mean the expected value of $\cos\gamma$ and I will denote that by $\E(\cos\gamma)$. Then \begin{align} & \E(\cos(\omega t + \psi) + \sin(\omega t + \gamma)) \\[8pt] = {} & \cos(\omega t)\E(\cos\psi) + \sin(\omega t)\E(\sin\psi) + \sin(\omega t)\E(\cos\gamma) + \cos(\omega t)\E(\sin\gamma). \end{align} This holds because $\omega t$, and hence all functions of $\omega t$ are constant, i.e. not random. If you need something like $\operatorname{cov}(Y(t),Y(s))$ you can similarly apply standard trigonometric identities.