Let $(x_1, x_2, x_3) \in \mathbb{R}^3$ be jointly normally distributed with mean $\mu = (\mu_1,\mu_2, \mu_3)$ and covariance matrix $\Sigma \in \mathbb{R}^{3 \times 3}$.
It is known that $y := x_1 + x_2+ x_3$ is also normally distributed with mean $\mu_y = \mu_1+\mu_2+\mu_3$ and variance $\sigma^2_y = \sum_{i=1}^3 \sum_{j=1}^3 \Sigma_{i,j}$.
My question is concerned with the conditional expectation and variance of $(x_1, x_2, x_3) \in \mathbb{R}^3$ given that the sum $y=x_1 + x_2+x_3 = a \in \mathbb{R}$ is fixed, i.e. $$ \mathbb{E}[(x_1, x_2, x_3) | x_1 + x_2+x_3 = a] . $$ What can be said about this setting? Is the conditional distribution still normal? How can the density be computed? Thanks!