LetX: Ω →Rbe a random variable defined on a probability space (Ω,P). ThenXis analmost surely constant random variableifand is furthermore a

- Pr(X = c) = 1,
constant random variableif

- X(ω) = c, ∀ω ∈ Ω.

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,