# Degenerate distribution

In mathematics, a **degenerate distribution** is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) 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. 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.

Cumulative distribution function CDF for k _{0}=0. The horizontal axis is x. | |||

Parameters | |||
---|---|---|---|

Support | |||

pmf | |||

CDF | |||

Mean | |||

Median | |||

Mode | |||

Variance | |||

Skewness | undefined | ||

Ex. kurtosis | undefined | ||

Entropy | |||

MGF | |||

CF |

In the case of a real-valued random variable, the degenerate distribution is localized at a point *k*_{0} on the real line. The probability mass function equals 1 at this point and 0 elsewhere.

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*_{0}, with infinite height there but area equal to 1.

The cumulative distribution function of the univariate degenerate distribution is:

## Constant random variable

In probability theory, a **constant random variable** is a discrete random variable that takes a constant value, regardless of any 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 *X*: Ω → **R** be a random variable defined on a probability space (Ω, *P*). Then *X* is an *almost surely constant random variable* if there exists such that

and is furthermore a *constant random variable* if

Note that a constant random variable is almost surely constant, but not necessarily *vice versa*, since if *X* is almost surely constant then there may exist γ ∈ Ω such that *X*(γ) ≠ *k*_{0} (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ *k*_{0}) = 0).

For practical purposes, the distinction between *X* being constant or almost surely constant is unimportant, since the cumulative distribution function *F*(*x*) of *X* does not depend on whether *X* is constant or 'merely' almost surely constant. In either case,

The function *F*(*x*) is a step function; in particular it is a translation of the Heaviside step function.

## Higher dimensions

Degeneracy of a multivariate distribution in *n* random variables arises when the support lies in a space of dimension less than *n*. 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*.

In general when one or more of *n* random variables are exactly linearly determined by the others, if the covariance matrix exists its determinant is 0, so it is positive semi-definite but not positive definite, and the joint probability distribution is degenerate.

Degeneracy can also occur even with non-zero covariance. For example, when scalar *X* is symmetrically distributed about 0 and *Y* is exactly given by *Y* = *X* ^{2}, all possible points (*x*, *y*) fall on the parabola *y = x* ^{2}, which is a one-dimensional subset of the two-dimensional space.