conditional expectation of gaussian random variable Let $(X_1, \ldots , X_n)$ be a gaussian random vector with zero mean and covariance matrix $B$.  Find the distribution of the random vector $E(X_1 \mid X_2, \ldots, X_n)$.  I am not sure how to proceed.  Thanks in advance.

You can find formulas for the conditional distributions from a Gaussian vector.
In particular, if $B=\begin{bmatrix} B_{11} & B_{12} \\ B_{12}^\top & B_{22} \end{bmatrix}$ where $B_{11}$ is $1 \times 1$ and $B_{22}$ is $(n-1) \times (n-1)$, then the conditional expectation is
$$E[X_1 \mid X_2 ,\ldots, X_n] = B_{12} B_{22}^{-1} \begin{bmatrix}X_2 \\ X_2 \\ \vdots \\ X_n \end{bmatrix}.$$
This is a linear transformation of the Gaussian vector $(X_2, \ldots, X_n)$ to a univariate random variable, so the distribution is univariate Gaussian. I'll leave it to you to compute the mean and variance of this univariate Gaussian.
