I know that if, for example, $(X, Y)$ is jointly normal, then any linear combination of them is normal.

Now my question is, is a collection of distinct pairwise sums jointly normal? If not, what kind of conditions are required for them to be jointly normal?


1 Answer 1


Affine transformations of multivariate normals are again multivariate normal. In your case,

$$\begin{pmatrix}X_1-X_4 \\ X_2-X_3 \\X_4-X_3 \end{pmatrix} = \underbrace{\begin{pmatrix} 1 & 0 & 0 & -1 \\0 & 1& -1 & 0 \\ 0 & 0 & -1 & 1 \end{pmatrix}}_{=:M} \underbrace{\begin{pmatrix}X_1 \\ X_2 \\ X_3 \\X_4 \end{pmatrix}}_{=:X} $$ is normal with mean $M\vec{\mu}$ and variance $M\Sigma M^t$, where $\vec\mu$ denotes the mean and $\Sigma$ the covariance matrix of $X$.


Your Answer

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

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