properties of conditional normal random vectors [duplicate]

Given the partitioning: $$X = \begin{pmatrix}{} x_1 \\ \ x_2 \end{pmatrix}$$ for a normal random vector with mean $$\mu = \begin{pmatrix}{} \mu_1 \\ \ \mu_2 \end{pmatrix}$$ and covariance matrix:

$$\Sigma = \begin{pmatrix}{} \Sigma_{11} & \Sigma_{12} \\ \ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$$ where $$\Sigma_{22}$$ is invertible. How can I understand the following mean,covariance of a distribution of a random variable $$X_1$$ under condition $$X_2 = x$$? I'm struggling with formula for both the mean and covariance which are given specifically as:

$$\mu_1 +\Sigma_{12}\Sigma_{22}^{-1}(x-\mu_2)$$

and

$$\Sigma_{11}- \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$$.

I'm just getting into multivariate statistics, so I would preferably just try to get some intuition behind these two formulas if possible.

• A great deal of intuition can come from studying the simplest (nontrivial) case: see our thread on this at stats.stackexchange.com/questions/71260. If that doesn't answer your question, could you please indicate what you might be looking for in addition?
– whuber
May 12 '21 at 15:53
• Yes, thank you. That solves my problem completely. May 13 '21 at 10:58