I have the following question given in Communication Systems by Dr Sanjay Sharma :- "Show that the random process $X(t) = A cos(\omega t + \theta)$ where $\theta$ is a random variable uniformly distributed in range $(0, 2 \pi )$ , is a wide sense stationary process." In the solution while calculating the mean, the author writes, $\mu _X(t) = \int_{-\infty}^{\infty}X f_X(x,t) dx$ and $f_X(x,t) = f_{\theta}(\theta) = \frac{1}{2\pi} U(0,2\pi)$. But while calculating mean of functions (before introducing random process) the book used the formula as $\mu _X = \int_{-\infty}^{\infty}x f_X(x) dx$. I am not able to get the meaning of the mean/expectation in random process (which one is random variable, which one is distribution function). according to me it should have been $\mu _X(t) = \int_{-\infty}^{\infty}\theta f_{\theta}(\theta) d\theta$.
PS. In Simon Haykins the formulae for mean is $\mu _X(t) = \int_{-\infty}^{\infty}x f_{X(t)}(x) dx$ that means the integration has to be performed wrt the same varible that is being multiplied to $f$.