Constant random variable - Definition and Overview

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

<math>\Pr(X = c) = 1,<math>

and is furthermore a constant random variable if

<math>X(\omega) = c, \quad \forall\omega \in \Omega.<math>

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,

<math>f(x) = \begin{cases}1, &x = c,\\0, &x \neq c.\end{cases} \qquad \mathrm{and} \qquad F(x) = \begin{cases}1, &x \geq c,\\0, &x < c.\end{cases}<math>

The function  F(x)  is a step function.