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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}

Sunhwa
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1 Answers1

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$E(\mathbf{y}|\mathbf{x})$, 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.

Mark L. Stone
  • 4,496
  • yes, but I am interested in what is the conditioning in the top and bottom elements. is it either on only the element that lines up to it, or is it both elements in the conditioning, ie eqn (1) or eqn (2)? – Sunhwa Mar 11 '18 at 00:02
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    The conditioning is on the combination of both elements, as with equation 2. – Mark L. Stone Mar 11 '18 at 00:04