Degenerate distribution

In mathematics, a degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die whose sides all show the same number. While this distribution does not appear random in the everyday sense of the word, it does satisfy the definition of random variable.

The degenerate distribution is localized at a point k₀ on the real line. The probability mass function is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Pr(X = c) = 1,

and is furthermore a constant random variable if

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 an event  γ ∈ Ω  such that  X(γ) ≠ c  (but then necessarily  P(γ) = 0).

For practical purposes, the distinction between  X  being constant or almost surely constant is unimportant, since the probability mass function  f(x)  and cumulative distribution function  F(x)  of  X  do not depend on whether  X  is constant or 'merely' almost surely constant. In either case,

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \begin{cases}1, &x = c,\\0, &x \neq c.\end{cases}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): F(x) = \begin{cases}1, &x \geq c,\\0, &x < c.\end{cases}

The function  F(x)  is a step function.

See also

Dirac delta function

it:Variabile casuale degenere

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