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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 -dimensional space endowed with the Lebesgue measure, any distribution concentrated on a -dimensional subspace with is a degenerate distribution on . 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 , this distribution is a Dirac measure in : it is the distribution of a deterministic random variable equal to with probability 1. This is a special case of a discrete distribution; its probability mass function equals 1 in and 0 everywhere else.
In the case of a real-valued random variable, the cumulative distribution function of the degenerate distribution localized in 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 variable
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 be a real-valued random variable defined on a probability space . Then 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 is almost surely constant then there may still exist such that . For practical purposes, the distinction between being constant or almost surely constant is unimportant, since these two situation correspond to the same degenerate distribution: the Dirac measure.
Almost surely constant random variables are independent of everything — that is, if <math>X</math> is almost surely constant, then for every event <math>A</math> and every measurable set <math>B \subset \mathbb{R}</math>,
<math display="block">\mathbb{P}(\{X \in B\} \cap A) = \mathbb{P}(X \in B)\times\mathbb{P}(A).</math>
In particular, an almost surely constant random variable is independent of itself. Moreover, this self-independence characterizes almost surely constant random variables: if a random variable is independent of itself, then it is almost surely constant.
Higher dimensions
Degeneracy of a multivariate distribution in n random variables arises when the support lies in a space of dimension less than n.