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{{Short description|Probability distribution}}
{{More citations needed|date=August 2021}}<!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion
of standards used for probability distribution articles such as this one. -->
{{Probability distribution|
  name       =Degenerate univariate|
  type       =mass|
  cdf_image  =[[Image:Degenerate.svg|325px|Plot of the degenerate distribution CDF for {{math|''a'' {{=}} 0}}]]<br /><small>CDF for {{math|''a'' {{=}} 0}}. The horizontal axis is {{mvar|x}}.</small>|
  parameters =<math>a \in (-\infty,\infty)\,</math>|
  support    =<math>\{a\}</math>|
  pdf        =<math>
    \begin{matrix}
    1 & \mbox{for }x=a \\ 0 & \mbox{elsewhere}
    \end{matrix}
    </math>|
  cdf        =<math>
    \begin{matrix}
    0 & \mbox{for }x<a \\1 & \mbox{for }x\ge a
    \end{matrix}
    </math>|
  mean       =<math>a\,</math>|
  median     =<math>a\,</math>|
  mode       =<math>a\,</math>|
  variance   =<math>0\,</math>|
  skewness   =[[0/0|undefined]]|
  kurtosis   =[[0/0|undefined]]|
  entropy    =<math>0\,</math>|
  mgf        =<math>e^{at}\,</math>|
  char       =<math>e^{iat}\,</math>|
  pgf        =<math>z^{a}\,</math>|
}}

In [[probability theory]], a '''degenerate distribution''' on a [[measure space]] <math>(E, \mathcal{A}, \mu)</math> is a [[probability distribution]] whose [[Support (measure theory)|support]] is a [[null set]] with respect to <math>\mu</math>. For instance, in the {{mvar|n}}-dimensional space {{math|ℝ{{sup|''n''}}}} endowed with the [[Lebesgue measure]], any distribution concentrated on a {{mvar|d}}-dimensional subspace with {{math|''d'' < ''n''}} is a degenerate distribution on {{math|ℝ{{sup|''n''}}}}.<ref name=":0">{{Cite web|title=Degenerate distribution - Encyclopedia of Mathematics|url=http://encyclopediaofmath.org/index.php?title=Degenerate_distribution|url-status=live|archive-url=https://web.archive.org/web/20201205021345/https://encyclopediaofmath.org/wiki/Degenerate_distribution|archive-date=5 December 2020|access-date=6 August 2021|website=encyclopediaofmath.org}}</ref> This is essentially the same notion as a [[singular measure|singular probability measure]], but the term ''degenerate'' is typically used when the distribution arises as a [[Convergence of random variables|limit]] of (non-degenerate) distributions.

When the support of a degenerate distribution consists of a single point {{mvar|a}}, this distribution is a '''[[Dirac measure]] in {{mvar|a}}''': it is the distribution of a deterministic random variable equal to {{mvar|a}} with probability&nbsp;1. This is a special case of a [[discrete distribution]]; its [[probability mass function]] equals 1 in {{mvar|a}} and 0 everywhere else.
 
In the case of a real-valued random variable, the [[cumulative distribution function]] of the degenerate distribution localized in {{mvar|a}} 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 distribution]]s whose [[variance]] goes to 0.

==Constant random variable==
A '''constant random variable''' is a [[discrete random variable]] that takes a [[Constant function|constant]] value, regardless of any [[event (probability theory)|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 {{math|''X'': Ω → ℝ}} be a real-valued random variable defined on a probability space {{math|(Ω, ℙ)}}.  Then {{mvar|X}} 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 {{mvar|X}} is almost surely constant then there may still exist {{math|γ ∈ Ω}} such that {{math|''X''(γ) ≠ a}}.

For practical purposes, the distinction between {{mvar|X}} being constant or almost surely constant is unimportant, since these two situation correspond to the same degenerate distribution: the Dirac measure.

==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}}

== References ==
{{Reflist}}{{ProbDistributions|miscellaneous}}

{{DEFAULTSORT:Degenerate Distribution}}
[[Category:Discrete distributions]]
[[Category:Types of probability distributions]]
[[Category:Infinitely divisible probability distributions]]