MLE for the mean and variance of a normal

Question

You observe n independent samples x1, \dots, xn from a normal distribution \mathcal{N}(\mu, \sigma^2) with both parameters unknown. Derive the maximum likelihood estimators of \mu and \sigma^2, and state which is biased.

Accepted Answer

Log-likelihood. \ell(\mu, \sigma^2) = -\frac{n}{2}\ln(2\pi) - \frac{n}{2}\ln\sigma^2 - \frac{1}{2\sigma^2}\sum{i=1}^n (xi - \mu)^2. Mean. Differentiate with respect to \mu: \frac{\partial \ell}{\partial \mu} = \frac{1}{\sigma^2}\sumi (xi - \mu) = 0 \implies \hat{\mu} = \bar{x}. Variance. Differentiate with respect to \sigma^2 and plug in \hat{\mu} = \bar{x}: \frac{\partial \ell}{\partial \sigma^2} = -\frac{n}{2\sigma^2} + \frac{1}{2\sigma^4}\sumi (xi - \bar{x})^2 = 0 \implies \hat{\sigma}^2 = \frac{1}{n}\sumi (xi - \bar{x})^2. Bias. The mean estimator is unbiased: \mathbb{E}[\bar{x}] = \mu, w…

MLE for the mean and variance of a normal

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