# Degenerate distribution

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

Parameters | [math]\displaystyle{ k_0 \in (-\infty,\infty)\, }[/math] | ||
---|---|---|---|

Support | [math]\displaystyle{ x=k_0\, }[/math] | ||

pmf | [math]\displaystyle{ \begin{matrix} 1 & \mbox{for }x=k_0 \\ 0 & \mbox{elsewhere} \end{matrix} }[/math] | ||

CDF | [math]\displaystyle{ \begin{matrix} 0 & \mbox{for }x\lt k_0 \\1 & \mbox{for }x\ge k_0 \end{matrix} }[/math] | ||

Mean | [math]\displaystyle{ k_0\, }[/math] | ||

Median | [math]\displaystyle{ k_0\, }[/math] | ||

Mode | [math]\displaystyle{ k_0\, }[/math] | ||

Variance | [math]\displaystyle{ 0\, }[/math] | ||

Skewness | undefined | ||

Ex. kurtosis | undefined | ||

Entropy | [math]\displaystyle{ 0\, }[/math] | ||

MGF | [math]\displaystyle{ e^{k_0t}\, }[/math] | ||

CF | [math]\displaystyle{ e^{ik_0t}\, }[/math] |

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.

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:

[math]\displaystyle{ F_{k_0}(x)=\left\{\begin{matrix} 1, & \mbox{if }x\ge k_0 \\ 0, & \mbox{if }x\lt k_0 \end{matrix}\right. }[/math]

## 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 [math]\displaystyle{ k_0 \in \mathbb{R} }[/math] such that

- [math]\displaystyle{ \Pr(X = k_0) = 1, }[/math]

and is furthermore a *constant random variable* if

- [math]\displaystyle{ X(\omega) = k_0, \quad \forall\omega \in \Omega. }[/math]

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,

- [math]\displaystyle{ F(x) = \begin{cases}1, &x \geq k_0,\\0, &x \lt k_0.\end{cases} }[/math]

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.

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Original source: https://en.wikipedia.org/wiki/ Degenerate distribution.
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