Conditional expectation of a vector

Question

Suppose we have two random vectors $X=(X_1,X_2)^T$ and $Y=(Y_1,\dots,Y_n)^T$. I wish to find a simple definition or formula for

$$ E_{X|Y=y}[X] $$

Intuitively, I think the following is correct:

$$ E_{X|Y=y}[X]=\begin{pmatrix}E_{X_1|Y=y}[X_1]\\E_{X_2|Y=y}[X_2]\end{pmatrix} \tag{1} $$

Is this right? I imagine another possible definition could be

$$ E_{X|Y=y}[X]=\begin{pmatrix}E_{X|Y=y}[X_1]\\E_{X|Y=y}[X_2]\end{pmatrix} \tag{2} $$

But I don't think (2) is correct. I say this because, in the case of discrete $X$, (3) is more intuitive to me than (4):

$$ \big[E_{X|Y=y}[X]\big]_1=\sum_{x_1}x_1p_{X_1|Y}(x_1|y)=\sum_{x_1}x_1P(X_1=x_1|Y=y) \tag{3} $$

$$ \big[E_{X|Y=y}[X]\big]_1=\sum_{x}x_1p_{X|Y}(x|y)=\sum_{x}x_1P(X=x|Y=y) \tag{4} $$

Answer 1

(1) is correct and intuitive; while (2) is also correct because you can calculate $E[X_1]$ using the density of $X_1$ or the joint density of $X_1$ and $X_2$, e.g. (in your notation)

$$\begin{align} E_{X|Y=y}[X_1] &= \sum_x{x_1P(X=x|Y=y)} \\ &= \sum_{x_1}\sum_{x_2}{x_1P(X_1=x_1,X_2=x_2|Y=y)} \\ &= \sum_{x_1}x_1\sum_{x_2}{P(X_1=x_1,X_2=x_2|Y=y)} \\ &= \sum_{x_1}{x_1}P(X_1=x_1|Y=y) \\ &= E_{X_1|Y=y}[X_1] \end{align}$$