$X=(X_1,X_2,X_3)^T\sim N_3(\mu,\Sigma).$ 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 $$\Sigma^* = S_{11}-S_{12}S_{22}^{-1}S_{21}^{-1} = \pmatrix{3,1\\1,4}-\pmatrix{2\\1}(3)^{-1}\pmatrix{2,1} = \pmatrix{5/3,1/3\\1/3,11/3}$$


1 Answer 1


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.

  • $\begingroup$ Thanks, I had a go using the formula but didn't understand how to implement it with this example. Did I do it correctly? $\endgroup$
    – CCZ23
    Sep 6, 2020 at 18:24
  • $\begingroup$ 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. $\endgroup$
    – gunes
    Sep 6, 2020 at 19:27
  • $\begingroup$ 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 $\endgroup$
    – CCZ23
    Sep 7, 2020 at 6:30
  • $\begingroup$ Yes, your reordering is correct $\endgroup$
    – gunes
    Sep 7, 2020 at 6:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.