Template:Short description Template:More citations needed Template:Probability distribution
In probability theory, a degenerate distribution on a measure space <math>(E, \mathcal{A}, \mu)</math> is a probability distribution whose support is a null set with respect to <math>\mu</math>. For instance, in the Template:Mvar-dimensional space Template:Math endowed with the Lebesgue measure, any distribution concentrated on a Template:Mvar-dimensional subspace with Template:Math is a degenerate distribution on Template:Math.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> This is essentially the same notion as a singular probability measure, but the term degenerate is typically used when the distribution arises as a limit of (non-degenerate) distributions.
When the support of a degenerate distribution consists of a single point Template:Mvar, this distribution is a Dirac measure in Template:Mvar: it is the distribution of a deterministic random variable equal to Template:Mvar with probability 1. This is a special case of a discrete distribution; its probability mass function equals 1 in Template:Mvar and 0 everywhere else.
In the case of a real-valued random variable, the cumulative distribution function of the degenerate distribution localized in Template:Mvar is <math display="block">F_{a}(x)=\left\{\begin{matrix} 1, & \mbox{if }x\ge a \\ 0, & \mbox{if }x<a \end{matrix}\right.</math> Such degenerate distributions often arise as limits of continuous distributions whose variance goes to 0.
Constant random variableEdit
A constant random variable is a discrete random variable that takes a constant value, regardless of any event that occurs. This is technically different from an almost surely constant random variable, which may take other values, but only on events with probability zero: Let Template:Math be a real-valued random variable defined on a probability space Template:Math. Then Template:Mvar is an almost surely constant random variable if there exists <math>a \in \mathbb{R}</math> such that <math display="block">\mathbb{P}(X = a) = 1,</math> and is furthermore a constant random variable if <math display="block">X(\omega) = a, \quad \forall\omega \in \Omega.</math> A constant random variable is almost surely constant, but the converse is not true, since if Template:Mvar is almost surely constant then there may still exist Template:Math such that Template:Math.
For practical purposes, the distinction between Template:Mvar being constant or almost surely constant is unimportant, since these two situation correspond to the same degenerate distribution: the Dirac measure.
Higher dimensionsEdit
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.Template:Citation needed
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" />Template:Verify source and its determinant is 0, so it is positive semi-definite but not positive definite, and the joint probability distribution is degenerate.Template:Citation needed
Degeneracy can also occur even with non-zero covariance. For example, when scalar X is symmetrically distributed about 0 and Y is exactly given by Y = X2, all possible points (x, y) fall on the parabola y = x2, which is a one-dimensional subset of the two-dimensional space.Template:Citation needed