Simple definition of elliptical distribution I'm looking for a simple explanation of what an elliptical distribution is and how it differs from a non-elliptical distribution. My knowledge of stats is very basic; Wikipedia was not much help on this topic (too technical for my level of understanding). Thanks.
 A: First, we consider central elliptically/spherically distributed random vectors.
A spherical distribution of a vector $X$ can be constructed by first choosing randomly the length of $X$. The distribution of this random length is described by that function $\Psi$ in the wikipedia article.
Then a random direction represented as a point on a sphere with radius 1 is choosen, statistically independendly from the length. $X$ will be that point multiplied with the chosen length.
The distribution becomes elliptical if instead a point on a more general ellipsoid is choosen (spheres are special ellipsoids). $X$ will again point in the direction of this point on the ellipsoid, but additionally to its already randomly choosen length, $X$ will also be stretched by the distance of the point on the ellipsoid to $0$.
Noncentral elliptical/spherical distributions are made by shifting this construction in the end.
See Cambanis, Huang, Simons (1981) for the theorem that proves this decomposition of a elliptical distribution in independend length and direction parts.
