Using two linear functions of a 3D random vector to find a plane in which it is concentrated

Question

Let us take three random normal variables and combine them into one which we call $X$. We know their means, variances, and covariances, and thus we can come up with a mean vector and a variance matrix: $$\xi = (0,0,0)^T \ \ \ \\ \Sigma = \begin{bmatrix} 1 & 1 & 0 \\1 & 4 & -3 \\0 & -3 & 3 \end{bmatrix}$$ I have to answer the following question. I have already answered it using a different method, but I cannot do it using the method described in the question:

Determine the distribution of \begin{pmatrix} X_1 - X_2 \\ X_3 \end{pmatrix} and use this information to determine a subspace $S \subset \mathbb{R}^3$ of dimension $2$ on which $X$ is concentrated with probability $1$.

The method I used was more by brute force and the answer I got was rather ugly.

Answer 1

Hint:

• What is the covariance of ${X_1-X_2 \choose X_3}$?

Further hints:

• $E[(X_1-X_2)^2] = E[X_1^2]-2E[X_1 X_2] + E[X_2^2]$
• $E[(X_1-X_2) X_3] = E[X_1 X_3]-E[X_2 X_3]$