I want to derive the distribution of a joint random vector $\pmatrix{ X \\ Y}$ conditioned on a sub-vector. Right now I focus on the 2-d case with real valued $X,Y$.
But I got confused by some articles. Say $\pmatrix{ X \\ Y}$ is jointly (e.g. normal) distributed, $A \in \mathcal{B}(\mathbb{R}^2)$ and $B \in \mathcal{B}(\mathbb{R})$. Say $A_1$ is the projection of $A$ onto the first coordinate and $A_2$ the projection on the second coordinate. Does the following equality hold while conditioning?
$$P(\pmatrix{ X \\ Y} \in A \mid Y \in B) = P(X \in A_1 \mid Y \in B) $$
and in general, is
$$ P(\pmatrix{ X \\ Y} \in A) = P(X \in A_1, Y \in A_2)$$ ?
I don't understand applying $P$ to a random variable. I would apply $P$ to an event and $E$ to a random variable.
Meaning? Maybe $$ \mathbb E\left[\pmatrix{ X \\ Y}\mid Y\right] = \pmatrix{ \mathbb E[X\mid Y] \\ \mathbb E[Y\mid Y]} $$