Say $X(t)$ is a stochastic process. Now when it says that the mean of the process, does it mean that the mean of $X(t)$.
Elaborating further, a process is an collection of random variables - {$X(t_1), X(t_2), X(t_3),...$} and so on. So is the mean of the process is the mean of individual random variables, provided they come from identical distribution. For instance, for standard normal case, the mean of the process is 0 and variance is 1.
But in case the random variables are dependent as can be calculated from their autocorrelation function, how we would go about calculating the mean and variance of the process?
Help would be greatly appreciated!!
The mean of a process means the means of the individual random variables. Formally, this can be defined as a function from the index set to real numbers: \begin{equation} m:T\rightarrow \mathbb{R}: m(t_i) = E(X(t_i)) \end{equation} If the $X(t)$:s happen to have the same mean, \begin{equation} \forall t\in T: m(t) = \mu, \end{equation} we can simplify by just saying that the mean of the process is $\mu$. In the general case with varying mean, there is no concept of 'mean' of the process that would be a real number, but instead mean of the process refers to the function $m(t)$ defined above.
For variance, the above description holds, just replace each 'mean' by 'variance'.
To describe the correlation structure, one can use the covariance function: \begin{equation} c: T\times T \rightarrow \mathbb{R}: c(t_1,t_2) = \textrm{Cov}(X(t_1),X(t_2)). \end{equation}