Sample Mean is Unbiased Estimator of Population Mean

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Theorem

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

Then:

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

is an unbiased estimator of $\mu$.


Proof

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

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


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We have:

\(\ds \expect {\bar X}\) \(=\) \(\ds \expect {\frac 1 n \sum_{i \mathop = 1}^n X_i}\)
\(\ds \) \(=\) \(\ds \frac 1 n \sum_{i \mathop = 1}^n \expect {X_i}\) Expectation is Linear
\(\ds \) \(=\) \(\ds \frac 1 n \sum_{i \mathop = 1}^n \mu\) as $\expect {X_i} = \mu$
\(\ds \) \(=\) \(\ds \frac n n \mu\) as $\ds \sum_{i \mathop = 1}^n 1 = n$
\(\ds \) \(=\) \(\ds \mu\)

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

$\blacksquare$


Sources

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