I was transitioning from a Random Vector to a Stochastic Process in a textbook.
The Expected Value of a Random Vector is: [A vector of the expected values of the Random Variable]
But the Expected Value of a Stochastic Process is given as:
E[X(t)] for any t belonging to T.
So am I correct in understanding that at any time t in a Stochastic Process, it's expected value is simply the Expected Value of the Variable at that particular time index "t"?
If it is the case, why is it any different from the Random Vector?
A (real) random vector can be viewed as either a random element of $\mathbb R^n$ or as a vector whose components are real-valued random variables. Similarly a stochastic process can be viewed as a random element of a function space or as a function of time where each value $X(t)$ is a random variable. Being that function spaces are vector spaces, no there's not really any major conceptual difference. (Although there are more technical difficulties when you have to deal with function spaces or an uncountable number of random variables.)