Degenerate distribution

Degenerate

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

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:

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

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.