Editing Degenerate distribution (section)

{{Short description|The probability distribution of a random variable which only takes a single value}}
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{{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.