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The following multivariate distribution is given:

$\left(\begin{array}{c}X_1 \\ X_2 \\ X_3\end{array}\right) \sim N_3\left(\left(\begin{array}{c}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{ccc}1 & 0 & -1\\ 0 & 3 & 0\\ -1 & 0 & 2\end{array}\right)\right)$

The goal here is to derive the conditional distribution of $X_3$ given $X_1=X_2=0$. This is difficult for me, because I do not know yet how to deal with two conditions. From what I found online is that:

The conditional distribution of $\mathbf{X}_1$ given known values for $\mathbf{X}_2=\mathbf{x}_2$ is a multivariate normal with:

\begin{align} \text{mean vector} & = \mathbf{\mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(x_2-\mu_2)}\\ \end{align}

\begin{align} \text{covariance matrix} & = \mathbf{\Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}} \end{align}

The problem is that the formula is given for only one condition, but my exercise has two conditions.

Could I please get feedback on this?

Tim

kjetil b halvorsen
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Tim
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    In this specific case, $X_3$ is independent of $X_2$ whether or not you condition on $X_1$ – Henry Sep 12 '22 at 19:23
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    the formula you have works for vectors; you can subsume your two scalar conditions as one vector condition; using your formula $\mathbf{X}_2 = [0,0]$. – John Madden Sep 12 '22 at 19:26
  • @Johnmadden How does that influence the partitioned covariance matrices $\Sigma_{12}$, $\Sigma_{22}$ in this specific problem?. Actually I don't really understand how these matrices are derived from the 'orginal' covariance matrix. Do you maybe have a link to literature of that? – Tim Sep 12 '22 at 19:51
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    $\Sigma_{1,2}$ contains 1 row, and 2 columns giving the covariance between $X_1$ and $X_2$ and then $X_1$ and $X_3$. $\Sigma_{2,2}$ is a 2x2 matrix with the variance of $X_2$ and $X_3$ on the diagonal and the covariance between them on the off diagonal. – John Madden Sep 12 '22 at 20:42
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    A special property of multivariate Normal distributions is that their conditional expectations are linear. That makes this a (simple) multiple regression problem in which the mean vector and covariance matrix are given. https://stats.stackexchange.com/a/108862/919 provides the reasoning and algorithms to do the needed computations. – whuber Sep 12 '22 at 21:37

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