Degenerate distribution

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a degenerate distribution (sometimes also Dirac distribution) is, according to some, a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> in a space with <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> only on a <a href="/facts/Manifold/rXPERJtu">manifold</a> of lower <a href="/facts/Dimension_(mathematics)/0ktmUyac">dimension</a>, and according to others 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 <a href="/facts/Dice/qBnylzwJ">die</a> whose sides all show the same number.[better source needed] This distribution satisfies the definition of "random variable" even though it does not appear <a href="/facts/Randomness/QJQBoEX5">random</a> in the everyday sense of the word; hence it is considered <a href="/facts/Degeneracy_(mathematics)/BqhqlXPH">degenerate</a>.
In the case of a real-valued random variable, the degenerate distribution is a <a href="/facts/One-point_distribution/EpsKKVRu">one-point distribution</a>, localized at a point k0 on the <a href="/facts/Real_line/6eq42xX5">real line</a>.[better source needed] The <a href="/facts/Probability_mass_function/LhurokRt">probability mass function</a> equals 1 at this point and 0 elsewhere.
The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose <a href="/facts/Variance/ULBJKXD1">variance</a> goes to 0 causing the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> to be a <a href="/facts/Delta_function/V8PjRP8T">delta function</a> at k0, with infinite height there but area equal to 1.
The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> of the univariate degenerate distribution is:

 
 
 
 
 F
 
 
 k
 
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 x
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 {
 
 
 
 
 1
 ,
 
 
 
 
 if 
 
 
 x
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 k
 
 0
 
 
 
 
 
 
 0
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 x
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 k
 
 0
 
 
 
 
 
 
 
 
 
 
 {\displaystyle F_{k_{0}}(x)=\left\{{\begin{matrix}1,&{\mbox{if }}x\geq k_{0}\\0,&{\mbox{if }}x<k_{0}\end{matrix}}\right.}

Degenerate distribution open-in-new

Degenerate distribution