題組內容

三、 A random sample \( X_1, X_2, \dots, X_n \) of size \( n \) is taken from \( N(\mu, \sigma^2) \), where the variance \( \theta = \sigma^2 \) is such that \( 0 < \theta < \infty \) and \( \mu \) is a known real number.


1. Show that the maximum likelihood estimator for \( \theta \) is \[ \hat{\theta} = \frac{\sum_{i=1}^{n} (X_i - \mu)^2}{n} . (10 \text{分}) \]