# What is the conditional covariance matrix of $(X_2,X_3)^T$ given $X_1$?

$$X=(X_1,X_2,X_3)^T\sim N_3(\mu,\Sigma).$$ Suppose $$X_1,...,X_{20}$$ are i.i.d. observations from $$X$$. The sample mean vector and the covariance matrix are then defined by

$$\bar{x} = (1,0,2)^T,\quad S=\pmatrix{3,2,1\\2,3,1\\1,1,4}$$

What is the conditional covariance matrix of $$(X_2,X_3)^T$$ given $$X_1$$?

The conditional distribution is given by

$$X_2,X_3 \mid X_1\sim N(\mu^*,\Sigma^*)$$ where $$\mu^*= \bar{x}_1+S_{12}S_{22}^{-1}(x_2-\mu_2) = \pmatrix{0\\2} + \pmatrix{2\\1}(3)^{-1}x_1= \pmatrix{2/3x_1\\2+1/3x_1}$$ and $$\Sigma^* = S_{11}-S_{12}S_{22}^{-1}S_{21}^{-1} = \pmatrix{3,1\\1,4}-\pmatrix{2\\1}(3)^{-1}\pmatrix{2,1} = \pmatrix{5/3,1/3\\1/3,11/3}$$

The iiid realisations of $$X$$ should use another notation than $$X_i$$ (e.g. something like $$X^i)$$ because it gets mixed up with $$X_1,X_2$$ and $$X_3$$ inside the vector $$X$$.
If you had known the true $$\mu,\Sigma$$, you could use the conditional distribution formulas given in wikipedia page to calculate it exactly. But, you have approximations for them, which are calculated from your random sample of size $$20$$, and you can calculate the resulting approximate distribution.
• In the formula, the second portion of the vector is given. So, you should first rearrange your mean and covariance matrix to reflect the ordering $(X_2,X_3,X_1)$, then let $\mathbf x_1=[X_2,X_3]^T$ and $\mathbf x_2=[X_1]$, and apply the remaining formulas. Sep 6, 2020 at 19:27
• how can i rearrange the covariance matrix in that order? Like this $\pmatrix{3,1,2\\1,4,1\\2,1,3}$ ? I think it works but just want to check Sep 7, 2020 at 6:30