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?
 A: 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*}
A: I'm going to start with a somewhat more general situation:
\begin{align}
U & \sim N(\mu, \Sigma) & & \mu\in\mathbb R^{p\times 1},\quad \Sigma\in\mathbb R^{p\times p} \\[10pt]
V & \sim N(\nu, T) & & \nu\in\mathbb R^{q\times 1}, \quad T\in\mathbb R^{q\times 1} \\[10pt]
\operatorname{cov}(U,V) & = \Upsilon & & \Upsilon\in\mathbb R^{p\times q} \\
\text{and so }\operatorname{cov}(V,U) & = \Upsilon^T & & \Upsilon^T\in\mathbb R^{q\times p} \\[10pt]
\text{or more tersely } \begin{bmatrix} U \\ V \end{bmatrix} & \sim N\left( \begin{bmatrix} \mu \\ \nu \end{bmatrix}, \begin{bmatrix} \Sigma & \Upsilon \\ \Upsilon^T & T \end{bmatrix} \right).
\end{align}
Then it can be shown that
$$
\operatorname{E}(V\mid U) = \nu + \Upsilon T^{-1} (U-\mu).
$$
You have a situation where $\Sigma=1,$ $\Upsilon = [\rho,\ldots,\rho],$ and $T$ is an $(n-1)\times(n-1)$ matrix whose every diagonal entry is $1$ and whose every off-diagonal entry is $\rho.$ Finding $T^{-1}$ is the next task. This is messier than I expected. I'm getting a $q\times q$ matrix with diagonal entries $\dfrac{n\rho(n-2)}{n(1-\rho)(1+(n-1)\rho)}$ and off-diagonal entries $\dfrac{-n\rho}{n(1-\rho)(1+(n-1)\rho)}.$ But you might want to check that.
If you have
$$
\begin{bmatrix} X \\ Y \end{bmatrix} \sim N\left( \begin{bmatrix} \mu \\ \nu \end{bmatrix}, \begin{bmatrix} \sigma^2 & \rho\sigma\tau \\ \rho\sigma\tau & \tau^2 \end{bmatrix} \right)
$$
where $\mu$ and $\nu$ are real numbers, then you have
$$
\operatorname{E}(Y\mid X) = \nu + \rho\tau\left( \frac{X-\mu}\sigma \right).
$$
To check all this, note that $\operatorname{E}(Y\mid X)$ is a function of $X$, and must have the same expected value as $Y,$ and must be equal to the expected value of $Y$ when $X$ is equal to the expected value of $X,$ and $Y-\operatorname{E}(Y\mid X)$ must be uncorrelated with $X.$
