spherical distribution

Spherical distribution

A random variable $X$ is spherically symmetric about $\mu$ if for every orthogonal matrix $Q$

$$Q(X-\mu )\sim X-\mu$$

Spherical distribution (rotational symmetry) $⟹$ isotropic distribution
So the distribution looks the same in every direction, as the name suggests.
The PDF thus depends only on the distance from the mean:

$$f(x)=g(\Vert x-\mu \Vert )$$

for some function $g$ and mean $\mu$.