MLE for a normal distribution's mean and variance

Question

You observe X1, \dots, Xn i.i.d. from \mathcal{N}(\mu, \sigma^2) with both parameters unknown. Derive the maximum likelihood estimators of \mu and \sigma^2. Is either biased?

Accepted Answer

Log-likelihood. \ell(\mu, \sigma^2) = -\frac{n}{2}\log(2\pi) - \frac{n}{2}\log \sigma^2 - \frac{1}{2\sigma^2}\sum{i=1}^n (Xi - \mu)^2. Maximize over \mu. Only the last term involves \mu: \frac{\partial \ell}{\partial \mu} = \frac{1}{\sigma^2}\sum{i=1}^n (Xi - \mu) = 0 \implies \hat{\mu} = \bar{X}, for any \sigma^2, and the second derivative -n/\sigma^2 is negative. The MLE of the mean is the sample mean. Maximize over \sigma^2. Substitute \hat{\mu} = \bar{X} and differentiate with respect to v = \sigma^2: \frac{\partial \ell}{\partial v} = -\frac{n}{2v} + \frac{1}{2v^2}\sum{i=1}^n (Xi - \bar{…

MLE for a normal distribution's mean and variance

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