(This is a follow up on another question Deriving the conditional distributions of a multivariate normal distribution.)

I struggle when I condition several variables on another. My question is how to find the distribution of: $(y_1, y_2|y_3)$ where \begin{align} Y &\sim MVN_3(μ, Σ) \\[5pt] \text{where } μ &= (0,0,0) \\[5pt] \text{ and covariance matrix is }Σ &= \begin{bmatrix}1 &\rho_{12} &\rho{13} \\ \rho_{12} &1 &\rho_{23} \\ \rho_{12} &\rho_{23} &1 \end{bmatrix} \end{align}

Can I use the same form as in the question above?

The other way would also be interesting: $(y_1|y_2, y_3)$...

  • $\begingroup$ Welcome to the site, @edi. I took the liberty of editing your post to use the $\LaTeX$-style markdown our site affords. Please ensure it still says what you want it to say. $\endgroup$ Apr 17 '17 at 12:19
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    $\begingroup$ The answer to your question is yes. The answer in the linked question implies a completely arbitrary decomposition of $m$ variables in a multivariate Normal distribution into $k$ variables and $m-k$ variables. You're certainly allowed to use that to decompose 3 into 2 and 1. $\endgroup$ Apr 17 '17 at 18:48