Editing Degenerate distribution (section)

==Higher dimensions==

Degeneracy of a [[multivariate distribution]] in ''n'' random variables arises when the support lies in a space of dimension less than ''n''.<ref name=":0" /> This occurs when at least one of the variables is a deterministic function of the others. For example, in the 2-variable case suppose that ''Y'' = ''aX + b'' for scalar random variables ''X'' and ''Y'' and scalar constants ''a'' ≠ 0 and ''b''; here knowing the value of one of ''X'' or ''Y'' gives exact knowledge of the value of the other. All the possible points (''x'', ''y'') fall on the one-dimensional line ''y = ax + b''.{{Citation needed|date=August 2021}} 

In general when one or more of ''n'' random variables are exactly linearly determined by the others, if the [[covariance matrix]] exists its rank is less than ''n<ref name=":0" />''{{Verify source|date=August 2021}} and its [[determinant]] is 0, so it is [[Positive semidefinite matrix|positive semi-definite]] but not positive definite, and the [[joint probability distribution]] is degenerate.{{Citation needed|date=August 2021}}

Degeneracy can also occur even with non-zero covariance. For example, when scalar ''X'' is [[symmetric distribution|symmetrically distributed]] about 0 and ''Y'' is exactly given by ''Y'' = ''X''<sup>2</sup>, all possible points (''x'', ''y'') fall on the parabola ''y = x''<sup>2</sup>, which is a one-dimensional subset of the two-dimensional space.{{Citation needed|date=August 2021}}