View source for Degenerate distribution

{{Short description|The probability distribution of a random variable which only takes a single value}}
{{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 k<sub>0</sub>=0]]<br /><small>CDF for k<sub>0</sub>=0. The horizontal axis is ''x''.</small>|
  parameters =<math>k_0 \in (-\infty,\infty)\,</math>|
  support    =<math>x=k_0\,</math>|
  pdf        =<math>
    \begin{matrix}
    1 & \mbox{for }x=k_0 \\ 0 & \mbox{elsewhere}
    \end{matrix}
    </math>|
  cdf        =<math>
    \begin{matrix}
    0 & \mbox{for }x<k_0 \\1 & \mbox{for }x\ge k_0
    \end{matrix}
    </math>|
  mean       =<math>k_0\,</math>|
  median     =<math>k_0\,</math>|
  mode       =<math>k_0\,</math>|
  variance   =<math>0\,</math>|
  skewness   =[[0/0|undefined]]|
  kurtosis   =[[0/0|undefined]]|
  entropy    =<math>0\,</math>|
  mgf        =<math>e^{k_0t}\,</math>|
  char       =<math>e^{ik_0t}\,</math>|
  pgf        =<math>1\,</math>|
}}

In [[mathematics]], a '''degenerate distribution''' (sometimes also '''Dirac distribution''') is, according to some,<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> a [[probability distribution]] in a space with [[support (mathematics)|support]] only on a [[manifold]] of lower [[dimension (mathematics)|dimension]], and according to others<ref name=":1"/> a distribution with support only at a single point. By the latter definition, it is a '''deterministic distribution''' and takes only a single value.  Examples include a two-headed coin and rolling a [[Dice|die]]<!-- Please do not change this to 'dice'. 'Die' is singular;'dice' is the plural of 'die'. --> whose sides all show the same number.<ref name=":1">{{Cite web|last=Stephanie|date=2016-07-14|title=Degenerate Distribution: Simple Definition & Examples|url=https://www.statisticshowto.com/degenerate-distribution/|access-date=2021-08-06|website=Statistics How To|language=en-US}}</ref>{{Better source needed|date=August 2021}} This distribution satisfies the definition of "random variable" even though it does not appear [[randomness|random]] in the everyday sense of the word; hence it is considered [[degeneracy (mathematics)|degenerate]].{{Citation needed|date=August 2021}}

In the case of a real-valued random variable, the degenerate distribution is a [[one-point distribution]], localized at a point ''k''<sub>0</sub> on the [[real line]].<ref name=":1" />{{Better source needed|date=August 2021}} The [[probability mass function]] equals 1 at this point and 0 elsewhere.{{Citation needed|date=August 2021}}

The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose [[variance]] goes to 0 causing the [[probability density function]] to be a [[delta function]] at ''k''<sub>0</sub>, with infinite height there but area equal to 1.{{Citation needed|date=August 2021}}

The [[cumulative distribution function]] of the univariate degenerate distribution is:

<math>F_{k_0}(x)=\left\{\begin{matrix} 1, & \mbox{if }x\ge k_0 \\ 0, & \mbox{if }x<k_0 \end{matrix}\right.</math>{{Citation needed|date=August 2021}}

==Constant random variable==
In [[probability theory]], 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.  Constant and almost surely constant random variables, which have a degenerate distribution, provide a way to deal with constant values in a probabilistic framework.

Let  &nbsp;''X'': Ω → '''R'''&nbsp; be a random variable defined on a probability space &nbsp;(Ω, ''P'').  Then &nbsp;''X''&nbsp; is an ''almost surely constant random variable'' if there exists <math> k_0 \in \mathbb{R} </math> such that
:<math>\Pr(X = k_0) = 1,</math>
and is furthermore a ''constant random variable'' if
:<math>X(\omega) = k_0, \quad \forall\omega \in \Omega.</math>

A constant random variable is almost surely constant, but not necessarily ''vice versa'', since if &nbsp;''X''&nbsp; is almost surely constant then there may exist &nbsp;γ ∈ Ω&nbsp; such that &nbsp;''X''(γ) ≠ ''k''<sub>0</sub>&nbsp; (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ ''k''<sub>0</sub>) = 0).

For practical purposes, the distinction between &nbsp;''X''&nbsp; being constant or almost surely constant is unimportant, since the [[cumulative distribution function]] &nbsp;''F''(''x'')&nbsp; of &nbsp;''X''&nbsp; does not depend on whether &nbsp;''X''&nbsp; is constant or 'merely' almost surely constant. In either case,
:<math>F(x) = \begin{cases}1, &x \geq k_0,\\0, &x < k_0.\end{cases}</math>
The function &nbsp;''F''(''x'')&nbsp; is a [[step function]]; in particular it is a [[translation (geometry)|translation]] of the [[Heaviside step function]].{{Citation needed|date=August 2021}}

==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]]