Sample Mean is Unbiased Estimator of Population Mean

Theorem
Let $X_1, X_2, \ldots, X_n$ be random samples from a population with mean $\mu$ and variance $\sigma^2$.

Then:


 * $\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$

is an unbiased estimator of $\mu$.

Proof
If $\bar X$ is an unbiased estimator of $\mu$, then:


 * $\displaystyle \expect {\bar X} = \mu$

We have:

So $\bar X$ is an unbiased estimator of $\mu$.