# Conditional distribution between jointly normal and univariate normal random variables

Let $$X = (X_1, X_2, X_3)^T\sim N_3(\mu, \Sigma)$$, where

$$\mu = \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix}, \Sigma = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \\ \end{pmatrix}$$

I want to find $$(X_1, X_2)^T|X_3$$.

I know that $$(X_1, X_2)^T \sim N_2(\begin{pmatrix} 1\\ 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 2 \\ \end{pmatrix})$$ and that $$X_3 \sim N_1(1, 2)$$.

My definition of conditional distribution of MVN however initially requires the distribution of $$(((X_1, X_2)^T)^T, X_3^T)^T$$, of which I am not sure.

Is there another way around this that I am missing? Thanks for your help.

• There may be a typo in your notes/book. By vector, you seem to mean column vector (e.g. your definition of $\mu$) and so $X= (X_1, X_2, X_3)^T$ is a column vector. Now, the usual meaning of $((X_1, X_2)^T, X_3)$ is just the row vector $(X_2, X_2, X_3)$ for which you have no definition of a MVN unless that $((X_1, X_2)^T, X_3)$ should read $((X_1, X_2)^T, X_3)^{\mathbf T}$ which becomes $(X_1, X_2, X_3)^T$ under the usual interpretation. – Dilip Sarwate Dec 23 '18 at 14:23
• Please decide where you want to ask your question. Cross posting should be avoided in general. – StubbornAtom Dec 23 '18 at 14:27
• Could you please explain what distinction you might be making between the distribution of $((X_1,X_2)^\prime,X_3)$ and $(X_1,X_2,X_3)$? – whuber Dec 23 '18 at 14:39
• @DilipSarwate You are right, I was confused over the use of the transpose. I actually require the distribution of $(((X_1, X_2)^T)^T, X_3 ^T)^T$ which is of course $(X_1, X_2, X_3)^T$ for which we have the distribution. – Jason Dec 23 '18 at 15:40
• @StubbornAtom Apologies and thanks for the heads up. Still a newbie. – Jason Dec 23 '18 at 15:40