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I'm just starting to study sampling distributions. At the very start, it proves that expectation of $\bar{X}$ is the population mean. I don't get the intuition of it. Also, the proof states that the value of Expectation of $X_i$ is $\mu$. How?

Could someone explain this to me?

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$X_i$ is a random variable sampled from $f_X(x)$. Its expected value is the population mean, i.e. $\mu$, because $\mu$ is the expected value of the distribution represented by density/mass function $f_X(x)$. It directly follows from the definition of expectation.

Similarly, if the expected value of each $X_i$ is $\mu$, what could be the expected value of the sample mean, $\bar{X}$, other than the population mean, $\mu$?

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