MLE: Normal Distribution

Suppose \(X_1,\dots,X_n\) are i.i.d. from a normal distribution \(\mathcal{N}(\mu,\sigma^2)\). What is the best prediction for \(\mu\) and \(\sigma^2\)?

Answer

The maximum likelihood estimators (MLEs) for the mean and variance of a normal distribution are:

\[ \hat{\mu}_{\text{MLE}} = \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i \]
\[ \hat{\sigma}^2_{\text{MLE}} = \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2 \]

Thus, the best prediction (in the MLE sense) for:

  • the population mean \(\mu\) is the sample mean \(\bar{X}\),

  • the population variance \(\sigma^2\) is the sample variance with denominator \(n\).

Notes and comments

Comment 1: Note that the variance estimator is biased, however, it is asymptotically unbiased (a general property of MLEs under standard regularity conditions).

Back to collection