What is the distribution of $(x_i, z_i)$?

Question

I suppose this is more of a question about the notation. Let $x_i$ be a $p$-dimensional random vector with $x_i \sim \mathcal{N}(\mu, BIB^T + I\sigma^2)$. Let $z_i$ also be a $p$-dimensional random vector with $z_i\sim \mathcal{N} (0,I)$.

I am asked to show that (as I quote) "the joint distribution of $(x_i, z_i)$ is a multivariate normal distribution", but what does $(x_i,z_i)$ stand for here?

Answer 1

This notation usually means a joint RV of $x_i$ and $z_i$. Imagine stacking the components of $z_i$ at the end of $x_i$.