If $X_1, \ldots, X_n$ are multivariate normal, how can we find $E[X_1 \mid X_2, \ldots, X_n]$? If $X_1, \ldots, X_n$ are multivariate normal with mean vector $\mathbf{0}$ , variance $1$, and common covariance $\rho$, how can we find $E[X_1 \mid X_2, \ldots, X_n]$?
Is there a way to do this with or without integrals?

We try the case that $n=2$. Note that $X_{1}$, $X_{2}$ are jointly normal with $X_{1}\sim X_{2}\sim N(0,1)$, and $\operatorname{corr}(X_1,X_2)=\rho\in[-1,1]$. There exist independent
normal random variables $Y_{1}$, $Y_{2}$ with $Y_{1}\sim Y_{2}\sim N(0,1)$
such that $X_{1}=\sqrt{1-\rho^{2}}Y_{1}+\rho Y_{2}$ and $X_{2}=Y_{2}$. Now 
\begin{eqnarray*}
E\left[X_{1}\mid X_{2}\right] & = & E\left[\sqrt{1-\rho^{2}}Y_{1}+\rho Y_{2}\mid Y_{2}\right]\\
 & = & \sqrt{1-\rho^{2}}E\left[Y_{1}\mid Y_{2}\right]+\rho E\left[Y_{2}\mid Y_{2}\right]\\
 & = & \sqrt{1-\rho^{2}}E\left[Y_{1}\right]+\rho Y_{2}\\
 & = & \rho X_{2}.
\end{eqnarray*}
