# Imputing missing values from multivariate Normal Distribution

Let's say my data of dimension $$3$$ has a multivariate normal distribution with known mean vector $$\mu$$ and covariance matrix $$\Sigma$$

I then observe a sample from the distribution but with $$1$$ value missing $$[ ?, x_2, x_3 ]$$

How can I find the most likely value of $$x_1$$ given $$x_2$$ and $$x_3$$?

Condition on $$(x_2, x_3)$$, $$P(x_1|x_2,x_3)$$ is still a Gaussian distribution.
$$\mu_1+\begin{bmatrix}\Sigma_{12} & \Sigma_{12} \end{bmatrix}\begin{bmatrix}\Sigma_{22} & \Sigma_{23} \\ \Sigma_{32} & \Sigma_{33}\end{bmatrix}^{-1}\begin{bmatrix} x_2 - \mu_2 \\ x_3-\mu_3 \end{bmatrix}.$$