Degenerate Distribution - 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 such that

and is furthermore a constant random variable if

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 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,

and

The function F(x) is a step function; in particular it is a translation of the Heaviside step function.