To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Degenerate distribution

From Wikipedia, the free encyclopedia

Degenerate univariate
Cumulative distribution function
Plot of the degenerate distribution CDF for k0=0

CDF for k0=0. The horizontal axis is x.
Parameters
Support
PMF
CDF
Mean
Median
Mode
Variance
Skewness undefined
Ex. kurtosis undefined
Entropy
MGF
CF

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 k0 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 k0, with infinite height there but area equal to 1.

The cumulative distribution function of the univariate degenerate distribution is:

YouTube Encyclopedic

  • 1/3
    Views:
    27 950
    11 567
    1 806
  • 19. Weak Law of Large Numbers
  • Mod-01 Lec-10 Special Distributions - I
  • Mod-07 Lec-20 Overview of Statistical Learning Theory; Empirical Risk Minimization

Transcription

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) ≠ k0  (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ k0) = 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.

This page was last edited on 15 August 2020, at 13:53
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.