Maximum Likelihood Estimator for Expectation of Normal Distribution is Unbiased

Theorem

Let $X$ be a continuous random variable with a normal distribution.

Let $\mu$ be the expectation of $X$.

The maximum likelihood estimator for $\mu$ with respect to $X$ is unbiased, and is the sample mean $\overline x$ of $X$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximum likelihood estimation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): maximum likelihood estimation