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 $y_2|y_1<a$, 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.