We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning ${\boldsymbol Y}$ into
$${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\ {\boldsymbol y}_2 \end{bmatrix}$$
Say that $f_Y$ is a probability density function for $Y$. What can we say about the probability density function $f_{y_1|y_2 = a}$ of conditional multivariate distribution $(y_1|y_2 = a)$?
Does this hold? :
$$f_{y_1|y_2 = a}(y_1) \stackrel{?}{=} {f_Y([y_1,y_2 = a]) \over \int_{y_1} f_Y([y_1,y_2 = a])}$$
Intuitively, I would assume that we could use $f_Y$ in the appropriate $y_2 = a$, and then normalize it to integrate to 1... Or is there some "gotcha" that the density scales differently in the space of $y_1$? I know that it the conditional distribution could be computed with the Schur complement but this could save computational time in cases when you don't actually need a density normalized to 1, which is my case.
PS: and then, probably, $\int_{y_1} f_Y([y_1,y_2 = a]) \stackrel{?}{=} f_{y_2}(y_2 = a)$, so perhaps the normalization would also be simple?
