# Deriving the conditional distribution of a multivariate normal, for inequalities

This question is slightly related to Deriving the conditional distributions of a multivariate normal distribution. In that question, the following situation was given.

If $$Y$$ follows a multivariate normal distribution, $$Y \sim N(\mu,\Sigma)$$ and you partition $$Y$$ into $$Y = [y_1,y_2]$$, how can you derive the resulting conditional distribution of $$y_2|y_1=a$$? In words, if you start from a multivariate normal distribution, and you fix some of the values($$y_1=a$$), what is the conditional distribution of the remaining elements of $$Y$$?

My question is, how can you arrive at the conditional distribution $$y_2|y_1, again for $$Y \sim N(\mu,\Sigma)$$ and $$Y = [y_1,y_2]$$? It seems to me that the approach used in the case $$y_2|y_1=a$$ does not work here.

We find $$f_{Y_1,Y_2}(y_1|y_2)$$, which is normal, with mean as a function of $$y_2$$, $$\mu_{y_1|y_2}(y_2)$$ and deviation, $$\sigma$$, as shown in the wiki page. Then, we can calculate $$P(Y_1. Using Bayes Rule, we'd have $$f_{Y_2|Y_1 which isn't normal any more (in general) due to the term $$\phi(y_2)$$. This was for the bivariate case (your variables seem to be), but can be generalized into multivariate similarly if you like. However, $$y_1$$ is always univariate since you compare it with a constant.