Degenerate distribution
The probability distribution of a random variable which only takes a single value / From Wikipedia, the free encyclopedia
In mathematics, a degenerate distribution is, according to some,[1] a probability distribution in a space with support only on a manifold of lower dimension, and according to others[2] 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 die whose sides all show the same number.[2][better source needed] This distribution satisfies the definition of "random variable" even though it does not appear random in the everyday sense of the word; hence it is considered degenerate.[citation needed]
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Cumulative distribution function CDF for k0=0. The horizontal axis is x. | |||
Parameters | |||
---|---|---|---|
Support | |||
PMF | |||
CDF | |||
Mean | |||
Median | |||
Mode | |||
Variance | |||
Skewness | undefined | ||
Excess kurtosis | undefined | ||
Entropy | |||
MGF | |||
CF |
In the case of a real-valued random variable, the degenerate distribution is a one-point distribution, localized at a point k0 on the real line.[2][better source needed] The probability mass function equals 1 at this point and 0 elsewhere.[citation needed]
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 k0, with infinite height there but area equal to 1.[citation needed]
The cumulative distribution function of the univariate degenerate distribution is: