Degenerate distribution - Wikipedia (FazBrowse)

CDF for {{math|''a'' {{=}} 0}}. The horizontal axis is {{mvar|x}}."},"parameters":{"wt":"<math>a \\in (-\\infty,\\infty)\\,</math>"},"support":{"wt":"<math>\\{a\\}</math>"},"pdf":{"wt":"<math>\n \\begin{matrix}\n 1 & \\mbox{for }x=a \\\\ 0 & \\mbox{elsewhere}\n \\end{matrix}\n </math>"},"cdf":{"wt":"<math>\n \\begin{matrix}\n 0 & \\mbox{for }x<a \\\\1 & \\mbox{for }x\\ge a\n \\end{matrix}\n </math>"},"mean":{"wt":"<math>a\\,</math>"},"median":{"wt":"<math>a\\,</math>"},"mode":{"wt":"<math>a\\,</math>"},"variance":{"wt":"<math>0\\,</math>"},"skewness":{"wt":"[[0/0|undefined]]"},"kurtosis":{"wt":"[[0/0|undefined]]"},"entropy":{"wt":"<math>0\\,</math>"},"mgf":{"wt":"<math>e^{at}\\,</math>"},"char":{"wt":"<math>e^{iat}\\,</math>"},"pgf":{"wt":"<math>z^{a}\\,</math>"},"1":{"wt":"\n"}},"i":0}}]}'>

Degenerate univariate

Cumulative distribution function

a = 0"}'>

Plot of the degenerate distribution CDF for a = 0

CDF for a = 0. The horizontal axis is {a}

eiat

μ. For instance, in the d-dimensional subspace with a, this distribution is a Dirac measure in a and 0 everywhere else.

In the case of a real-valued random variable, the cumulative distribution function of the degenerate distribution localized in X: Ω → ℝ be a real-valued random variable defined on a probability space a∈ℝ such that X is almost surely constant then there may still exist X being constant or almost surely constant is unimportant, since these two situation correspond to the same degenerate distribution: the Dirac measure.

Almost surely constant random variables are independent of everything — that is, if B⊂ℝ, and its determinant is 0, so it is positive semi-definite but not positive definite, and the joint probability distribution is degenerate.
1 2 3 "Degenerate distribution - Encyclopedia of Mathematics". encyclopediaofmath.org. Archived from the original on 5 December 2020. Retrieved 6 August 2021.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Continuous binomial Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Power function Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized hyperbolic Generalized logistic (logistic-beta) Generalized normal Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's SU Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
Multinomial
- Dirichlet
- Negative
Continuous:
Dirichlet
- Generalized
Multivariate Laplace
Multivariate normal
Multivariate stable
Multivariate t
Normal-gamma
- Inverse
Matrix-valued:
LKJ
Matrix beta
Matrix F
Matrix normal
Matrix t
Matrix gamma
- Inverse
Wishart
- Normal
- Inverse
- Normal-inverse
- Complex
Uniform distribution on a Stiefel manifold

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: Dirac delta function
Singular: Cantor

Families

View original on Wikipedia | Back | FazBrowse Home