# Constant random variable

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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

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 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,

The function *F*(*x*) is a step function.

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