Conditioning on vector in Expectation

Question

If

$$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$

and

$$ \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} $$

Then what does

$$ E(\mathbf{y}|\mathbf{x}) = E \left( \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \bigg| \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)$$

look like, is it

\begin{equation}\tag{1} E \left( \begin{bmatrix} y_1 | x_1 \\ y_2 | x_2 \end{bmatrix}\right) \end{equation}

or

\begin{equation}\tag{2} E \left( \begin{bmatrix} y_1 | x_1, x_2 \\ y_2 | x_1, x_2 \end{bmatrix}\right) \end{equation}

Answer 1

$E(\mathbf{y}|\mathbf{x})$, without expanding it out, is a common way of writing it.

And then perhaps $E \left( \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \bigg| \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)$ would be the next most common way, but not very common at all.