Mean and Variance Through Maximum Likelihood

# Mean and Variance Through Maximum Likelihood Suppose observations `x1, ..., xn` are independent draws from a normal distribution with unknown mean `mu` and unknown variance `sigma_squared`. Derive the maximum-likelihood estimators for both parameters. Explain why the variance estimator uses `n` in its denominator, how it differs from the common unbiased sample-variance estimator, and what changes when the model assumptions are poor. ### Constraints & Assumptions - `n` is positive and every observation is finite. - The likelihood model is normal with independent, identically distributed observations. - Discuss the boundary case in which all observations are equal. - Distinguish maximum likelihood from unbiasedness. ### Clarifying Questions to Ask - Are both parameters unknown and estimated from the same sample? - Is the requested variance the maximum-likelihood estimate or an unbiased estimate? - Should the derivation optimize variance or standard deviation? ```hint Work with the log-likelihood After taking logs, separate the terms that depend on the mean from those that depend on the variance and differentiate each parameter. ``` ### What a Strong Answer Covers - The normal log-likelihood up to an additive constant - The first-order condition yielding the sample mean - The maximum-likelihood variance with denominator `n` - The distinction from Bessel's correction and the finite-sample bias - Boundary behavior, assumptions, and robustness concerns ### Follow-up Questions 1. Why does replacing `n` with `n - 1` remove bias under the normal iid model? 2. What estimator would you consider if large outliers make the normal likelihood implausible? 3. How does a known mean change the variance derivation?