Proof of Expectation Formula

Question

Prove that $E(X) = \mu$, where $X$ is the distribution of the sample mean and $\mu$ is the population mean. That is, the expected value of the sample mean $X$ is equivalent to the population mean. What is the mathematical proof of this?

Answer 1

Let $X_1,X_2,...,X_n$ be the sample from the population

Let $\overline{X}$ be the sample mean, let $\mu$ be the population mean. Then we have that

$E(\overline{X})=E(\frac{1}{n}\sum_{i=1}^n X_i) = \frac{1}{n}\sum_{i=1}^nE( X_i)=\frac{1}{n}nE(X_1)=\frac{1}{n}n\mu=\mu$

as each $E(X_i)$ is equal to one another as it is the average from the population as a whole.

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