I know that the best estimator is $g(x)=E\{Y|X=x\}$ and the conditional density for jointly Gaussian random variables is known to be Gaussian with mean and variance given by
\begin{equation} \eta_{Y|x} = \eta_{y} + \frac{\rho_{xy} \sigma_{y}( x - \eta_{y} )}{ \sigma_{x} } \end{equation} and \begin{equation} \sigma_{Y|x}^{2} = \sigma_{Y}^{2} ( 1 - \rho_{XY}^{2} ) \end{equation}
I am not sure how I can get that the optimal g(x) for this Gaussian case the conditional is the conditional mean:
\begin{equation} g(x) = \eta_{x} + \frac{ \rho_{xy} \sigma_{y} (x - \eta_{y}) }{ \sigma_{x} }. \end{equation}