Consider a sequence of random variables \(\left\{\left\{X_{i j}ight\}_{j=1}^{k}ight\}_{i=1}^{n}\) that are assumed to be mutually independent, each having a \(\mathrm{N}\left(\mu_{i}, \thetaight)\) distribution for \(i=\) \(1, \ldots, n\). Prove that the maximum likelihood estimators of \(\mu_{i}\) and \(\theta\) are

\[\hat{\mu}_{i}=\bar{X}_{i}=k^{-1} \sum_{j=1}^{k} X_{i j}\]

for each \(i=1, \ldots, n\) and

\[\hat{\theta}_{n}=(n k)^{-1} \sum_{i=1}^{n} \sum_{j=1}^{k}\left(X_{i j}-\bar{X}_{i}ight)^{2}\]