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I am new to time-series analysis and I found myself getting confused by the most fundamental concepts.

A stochastic process is a collection of random variables $X_t$ for $t=1,\dots,n$ where each $X_t$ is a random variable with its own probability distribution. A stochastic process can generate an infinite collection of realizations, and we call this collection an ensemble. (And an observed time series is 1 realization from that ensemble.) Am I correct in this understanding?

So when we say the expected value of a time series $E(X_t)=\mu_X(t)$, do we mean the expected value of a particular $X_t$ from the sequence, let's say $X_3$, which we can find by calculating the weighted average/integral of the probability distribution of $X_3$? And is it correct to assume that each $X_t$ in the sequence would have a different expected value since they most likely will not have the same probability distribution?

Similarly, a white noise $\{w_1,\dots,w_n\}$ is defined as a collection of random variables with mean 0 and s.d. $\sigma_w$. Does this mean each $w_i\in\{w_1,\dots,w_n\}$ is a random variable that has a probability distribution with mean 0 and every $w_i$ have the same s.d. $\sigma_w$?

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You're correct in your understanding. A realization is an observation from the infinite ensemble, each $X_t$ has its own distribution and at the same time sets of $X_t$ have joint distributions. The mean process $\mu_x(t)$ describes the mean of a specific $X_t$, e.g. $\mu_x(3)$ is the mean of $X_3$, which can be found by the distribution of $X_3$. Each $X_t$ may or may not have different means or other statistics based on the stationarity properties of the underlying random process. For example, if the process is mean-stationary, $\mu_x(t)=\mu$, i.e. constant, and each $X_t$ will have the same mean. But yes, in general, means are different. White noise is an example of a mean and covariance stationary process.

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