Editing Maxwell's theorem (section)

== Equivalent statements ==
If the probability distribution of a [[vector space|vector]]-valued [[random variable]] ''X'' = ( ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> )<sup>''T''</sup> is the same as the distribution of ''GX'' for every ''n''×''n'' [[orthogonal matrix]] ''G'' and the components are [[statistical independence|independent]], then the components ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[normal distribution|normally distributed]] with [[expected value]] 0 and all have the same [[variance]].  This theorem is one of many [[characterization (mathematics)|characterizations]] of the normal distribution.

The only rotationally invariant probability distributions on '''R'''<sup>''n''</sup> that have independent components are [[multivariate normal distribution]]s with [[expected value]] '''0''' and [[variance]] ''σ''<sup>2</sup>''I''<sub>''n''</sub>, (where ''I''<sub>''n''</sub> = the ''n''×''n'' identity matrix), for some positive number ''σ''<sup>2</sup>.