A random vector $X \in \mathbb{R}^n$ is isotropic if $\mathbb{E}XX^T = I\_n$. However isotropic random vectors don't have the property of isotropy. See [1](https://math.stackexchange.com/questions/2569824/if-x-is-isotropic-random-vector-then-is-the-centered-random-vector-x-ex-als?rq=1). So why are they called isotropic?


Similarly a random vector $X$ is anisotropic if $X = AY$ where $Y$ is isotropic. Although this might align more with the meaning of anisotropy, I'm not sure why these words are chosen.


Edit: Definition of [isotropy](https://en.wikipedia.org/wiki/Isotropy)



Why are isotropic random vectors called isotropic if they aren't?
This is a contamination quite common in probability when properties of distributions are instead attributed to the associated random objects. Strictly speaking one should talk about isotropic (i.e., rotation invariant) measures or distributions rather than vectors.


Yet another more recent example of this contamination is provided by so called "invariant random subgroups" which are in reality probability measures on the space of subgroups invariant with respect to the group action by conjugations. I wouldn't be surprised if one day probabilists begin to talk about "invariant random points" instead of invariant measures of a dynamical system.


