Let X: Ω → R be a random variable defined on a probability space (Ω, P). Then X is an almost surely constant random variable ifand is furthermore a constant random variable if
- Pr(X = c) = 1,
- 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,