Degenerate distribution

Degenerate
Cumulative distribution function CDF for k₀=0. The horizontal axis is the index i of k_i.
Parameters	$k_0 \in (-\infty,\infty)\,$
Support	$k=k_0\,$
pmf	δ $({x-k_0\,})$
CDF	$\begin{matrix} 0 & \mbox{for }k<k_0 \\1 & \mbox{for }k\ge k_0 \end{matrix}$
Mean	$k_0\,$
Median	$k_0\,$
Mode	$k_0\,$
Variance	$0\,$
Skewness	undefined
Ex. kurtosis	undefined
Entropy	$0\,$
MGF	$e^{k_0t}\,$
CF	$e^{ik_0t}\,$

In mathematics, a degenerate distribution or deterministic distribution is the probability distribution of a random variable which only takes 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 k₀ on the real line. The probability mass function equals 1 at this point and 0 elsewhere.

The 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.

The cumulative distribution function of the degenerate distribution is:

$F(k;k_0)=\left\{\begin{matrix} 1, & \mbox{if }k\ge k_0 \\ 0, & \mbox{if }k<k_0 \end{matrix}\right.$

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 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 $c \in \mathbb{R}$ such that

\Pr(X = c) = 1,

and is furthermore a constant random variable if

X(\omega) = c, \quad \forall\omega \in \Omega.

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) ≠ c (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ c) = 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 this case,

F(x) = \begin{cases}1, &x \geq c,\\0, &x < c.\end{cases}

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

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace Asymmetric Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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