Random Variables: Neither Random nor Variables

Random variables are a cornerstone of statistics, yet their name is a notorious misnomer. They aren't "random" in the colloquial sense, nor are they "variables" like those in algebra. What are they?

If you’ve studied probability or statistics you’ve encountered random variables—a concept so fundamental that its misleading name often goes unquestioned.

The truth? Random variables are neither “random” nor “variables” in the usual sense. So why do we call them that? And what are they, really?

Why are “random variables” not random?

At first glance, the term suggests unpredictability—a number that changes haphazardly. But mathematically, a random variable is not inherently random. Instead, it’s a function that assigns numerical values to outcomes of a probabilistic experiment.

For example:

Roll a die. The outcome is random (1, 2, …, 6). Then define a random variable X as:

\(X= \begin{cases} 1 & \text{if the roll is even,}\\ 0 & \text{if the roll is odd.} \end{cases}\)

Here, X is a fixed rule (a function), not something that fluctuates on its own. The randomness comes from the experiment (the die roll), not the variable itself.

Why are “random variables” not variables?

In algebra, a variable, like x in y = 2 ⋅ x + 3, represents an unknown or changing quantity. But a random variable is not an algebraic variable—it’s a measureable function from a sample space to real numbers.

Formal definition

A random variable X is a function:

\(X:\Omega\rightarrow\mathbb{R}\)

where Ω is the sample space (set of all possible outcomes) and ℝ is the set of real numbers.

Once the outcome ω ∈ Ω is observed, X(ω) is deterministic—no longer "variable."

So why call them “random variables”?

The name is a historical artifact. Early probabilists (like Laplace and Kolmogorov) needed a way to formalize uncertain quantities. The term "variable" stuck because it feels intuitive (we model fluctuating quantities) and it simplifies notation (writing P(X = x) is cleaner than P({ω : X(ω) = x}) ).

But in rigorous probability theory, random variables are static mappings, not fluctuating values.

So, why does this even matter?

Misunderstanding random variables may cause confusion in:

1. Conditioning

\(P(X|Y)\)

P(X | Y) isn’t updating a "variable" but a function’s behavior under constraints.

2. Expectation & Variance

\(\mathbb{E}[X], \text{Var}(X)\)

These are properties of the function’s output distribution, not of a changing value.

3. Stochastic Processes

\(\{X_t\}_{t \in \mathcal{T}}\)

A "time-varying random variable" is really a family of functions indexed by time.

A better name?

Some argue for "random function" or "measurable function," but tradition prevails. The key takeaway?

Random variables don’t vary—they translate randomness into numbers.

So, next time you see X~𝒩(0,1) remember: X is just a rule saying, "If the world is in state ω, output a number based on this distribution."

The magic isn’t in the name—it’s in the math.