Deriving the conditional distributions of a multivariate normal distribution

Overall, the intuition behind the conditional distribution of a bivariate normal is that even though the two variables are correlated in the joint distribution, they can still be treated as independent when you’re looking at the distribution of one variable, given a fixed value of the other variable. This allows you to use the normal distribution, which is a well-understood and widely-used probability distribution, to model the conditional distribution of a bivariate normal.

More generally,

Assume we know there’s a theorem that says all conditional distributions of a multivariate normal distribution are normal.