Degenerate distribution

From Wikipedia, the free encyclopedia

It has been suggested that this article or section be merged with Constant random variable. (Discuss)

Degenerate
Probability mass function PMF for k₀=0. The horizontal axis is the index k. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function CDF for k₀=0. The horizontal axis is the index k.
Parameters	$k_0 \in (-\infty,\infty)\,$
Support	$k=k_0\,$
Probability mass function (pmf)	$\begin{matrix} 1 & \mbox{for }k=k_0 \\0 & \mbox{otherwise } \end{matrix}$
Cumulative distribution function (cdf)	$\begin{matrix} 0 & \mbox{for }k<k_0 \\1 & \mbox{for }k>k_0 \end{matrix}$
Mean	$k_0\,$
Median	N/A
Mode	$k_0\,$
Variance	$0\,$
Skewness	$0\,$
Excess kurtosis	$0\,$
Entropy	$0\,$
Moment-generating function (mgf)	$e^{k_0t}\,$
Characteristic function	$e^{ik_0t}\,$

In mathematics, a degenerate distribution is the probability distribution of a discrete random variable that assigns all of the probability, i.e. probability 1, to a single number, a single point, or otherwise to just one outcome of a random experiment. Examples are a two-headed coin, a die that always comes up six. This does not sound very random, but it satisfies the definition of random variable.

The degenerate distribution is localized at a point $k 0$ in the real line. The probability mass function is given by:

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

The cumulative distribution function of the degenerate distribution is then:

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

There can be some ambiguity in the value of the cumulative distribution function at $k = k 0$ . In the above case the convention $F (k 0; k 0) = 1$ has been chosen.

[edit] Status of its PDF

As a discrete distribution, the degenerate distribution does not have a density.

The degenerate distribution of a continuous variable is described by the Dirac delta function.

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse Gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular

Retrieved from "http://en.wikipedia.org../../../d/e/g/Degenerate_distribution.html"

Categories: Articles to be merged since December 2006 | Discrete distributions

Degenerate distribution

From Wikipedia, the free encyclopedia

[edit] Status of its PDF

Views

Navigation

interaction

Search

In other languages