$\bar{X}$ versus $\mathbb{E}(\bar{X})$? I was not able to find this question here, so I am going to ask this:


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*What is the difference between $\mathbb{E}(\bar{X})$ (expected value of $X$ bar) and the actual $\bar{X}$?
I am very confused about these two concepts. How come $\bar{X}$ is one of the estimators of normal distribution (the other one being $S^2$) and then what is the point of $\mathbb{E}(\bar{X})$?

*Another question that I have is, what is the relationship between $\bar{X}$ and $E(\widehat{\mu})$
? I understand that mu is a true mean, but then what is $E(\widehat{\mu})$?
I would appreciate any explanation on these concepts and since I'm only a beginner in math stats, I am struggling with notation, so any simplified explanation will be much appreciated!
And then, I just discovered that there is also an expected value of $\sigma^2$ which, why lie, completely blew my mind! So I suspect I'm struggling with the whole concept of expected value and how it relates to the the population, sample distribution, and sampling distribution, so I would definitely appreciate an explanation of expected value of $\bar{X}$ in light of this.
 A: First of all, $\bar{X}$ is not an estimator of the Normal distribution. The true mean $\mu$ (as well as the true variance $\sigma^2$) is a parameter of the Normal distribution. $\bar{X}$ is an estimator for $\mu$ and this distinction is extremely important.
It sounds like that your confusion stems from not understanding that estimators themselves have distributions.
Suppose you have a true mean $\mu$ and you try to estimate what that true $\mu$ is. You do this by getting data through an experiment and compute $\bar{X}$. But $\bar{X}$ may not necessarily equal $\mu$; in other words, there is a variance to $\bar{X}$. So although the true $\mu$ may be $10$, your $\bar{X}$ may be $10.1$, or $9.7$ or some other value. So you should think of $\mathbb{E}(\bar{X})$ as the mean of the estimator of the mean. So since $\mathbb{E}(\bar{X}) = \mu$, we know that our realization of the sample mean (the $\bar{X}$) will be drawn from a distribution around $\mu$.
It would be fantastic to plug in $\mathbb{E}(\bar{X})$ to the Normal distribution since it equals $\mu$, but we don't know what $\mathbb{E}(\bar{X})$ is since the data can only tell us the realization of $\bar{X}$.
As to your second question, sometimes people write $\hat{\mu}$ is mean $\bar{X}$. The hat denotes that $\hat{\mu}$ is an estimator for $\mu$, which is what $\bar{X}$ is.
Like above, the $\mathbb{E}(\hat{\sigma}^2)$ (notice the hat) is the expected value of the estimator $\hat{\sigma}^2$, which is an estimator for the parameter $\sigma^2$ as you wrote above.
