Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a \(\mathrm{N}\left(\theta, \sigma^{2}ight)\) distribution conditional on \(\theta\), where \(\theta\) has a \(\mathrm{N}\left(\lambda, \tau^{2}ight)\) distribution where \(\sigma^{2}, \lambda\) and \(\tau^{2}\) are known. Prove that the posterior distribution of \(\theta\) is \(\mathrm{N}\left(\tilde{\theta}_{n}, \tilde{\sigma}_{n}^{2}ight)\) where

\[\tilde{\theta}_{n}=\frac{\tau^{2} \bar{x}_{n}+n^{-1} \sigma^{2} \lambda}{\tau^{2}+n^{-1} \sigma^{2}}\]

and

\[\tilde{\sigma}_{n}^{2}=\frac{\sigma^{2} \tau^{2}}{n \tau^{2}+\sigma^{2}}\]