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I have been trying to understand the methodology and application of conditional distributions in a multivariate normal setting as outlined in this Wikipedia page. In the context of that page, if we know the exact value of $x_2=a$, we can derive the conditional variance covariance matrix for $x_1$.

Now, e.g., if I have 3 variables $x_1$,$x_2$ and $x_3$ in a MVN distribution, and I get the conditional distribution of $x_1$ and $x_2$ given that I know $x_3$ (either as a single value or as a distribution?), what information is this derived conditional distribution giving me? What are the applications of this distribution? Can I use this conditional distribution to make out-of-sample predictions or at least predictions at values within the range of the $x_3$ not present in the original data.

I would appreciate if someone could point me to some examples of simple applications or literature. I'm not too technical but will do my best to read through and understand.

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