Let $Y\sim U(0,2\pi)$ be a random variable and consider a stochastic process $\{X_t,t\in\mathbb R\},$ where $X_t=a\sin(\omega t+Y)$ with $a>0,\omega>0$. Compute expected value and variance of this process.
I used $E(X_t)=a\sin(\omega t)E(\cos Y)+a\cos(\omega t)E(\sin Y),$ then separately computed densities of transformed variables $\cos Y,\sin Y$ and eventually $E(\cos Y)=E(\sin Y)=0.$ This also yields $\operatorname{Var}(X_t)=E((X_t-E(X_t))^2)=E(X_t^2)=\ldots$ (I didn't finish this because I'm lazy :)
But in retrospect, this seems to be clumsy. So I want to ask, is there a different way to determine the required moments?