# Why was the term random "variable" applied to a mapping?

I think I'm correct in saying a random variable is a mapping from the sample space to the real line (or more generally to $$\mathbb{R}^n$$. If I'm right then random variable seems a very odd way for a mathematician to describe it: it is neither random (a perfectly fixed mapping) nor a variable. Yet I have to believe that very smart people came up with this. I'm wondering who and what was the thinking behind the label.

• You are not correct in saying that. Identification of random variables with measurable maps is a device in Kolmogorov's formalization of probability. They are no more that than numbers are sets they are identified with in set theory, and working probability theorists do no think of them that way even today, it is just a technical artifice in proofs. As for pre-Kolmogorov genesis of the concept it is unclear, see Who introduced random variables into probability? and Who coined the term random variable? Dec 29 '21 at 5:34
• Thank you for responding. I didn't mention Kolmogorov and I didn't intend to imply any connection between his development of the axioms of probability theory and random variables. Dec 29 '21 at 6:57
• Sorry, somehow my comment uploaded before i'd finished writing it. Picking up where my previous comment ended: My question is simple: where did the term originate and why, given that it seems to to be an odd label. I don't think I'm wrong in my opening proposition, so I think my question still stands. If my opening proposition is wrong then I apologize and withdraw. Dec 29 '21 at 7:25
• @Coinfold. BTW, I'd read the two references you gave but they didn't seem to answer my question, hence I decided to post. I didn't want you to think I hadn't seen them. Dec 29 '21 at 7:29
• The only objectionable part is that random variable is a map, rather than that it is represented by one formally, but such oblivious shorthand is commonplace in the literature. However, that is not how random variables are treated informally, and the reason why the name does not match the technical definition, which it predates. But the question about its origin is interesting, hopefully someone can answer this time. Dec 29 '21 at 7:41

It is easy to name some of these "smart people".

Andrei Kolmogorov proposed a mathematical model for probability in his book Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of probability theory), Berlin: Julius Springer, (1933). This model is commonly accepted nowadays. This was a final step of a long development (Emile Borel, Francesco Cantelli, Felix Hausdorff, Maurice Frechet, Antoni Lomnicki, Evgeny Slutsky, A. A. Markov and many others).

There were several competing mathematical models, most notable of them due to

Sergei Bernstein, Theory of probability, Kharkov, 1911,

Hugo Steinhaus, Les probabilités dénombrables et leur rapport à la théorie de la mesure, Fundamenta math. 4, 286-310 (1923), and

Richard von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, J. Springer (1928).

Gradually the mathematical advantages of Kolmogorov axiomatization were recognized, and all other approaches were forgotten.

"Random variable" is an old term whose use started before these axiomatizations. It was thought as a "variable" (for example, a number) that "depends on chance". The last expression "depends on chance" required a mathematical explanation of "chance", and that the random variable "depends on it" means that it is really a "function of something". Axiomatization of Steinhaus is the closest to the one by Kolmogorov (the difference is that he used the newly invented Lebesgue measure to axiomatize the "chance", while Kolmogorov proposed to consider an abstract measure, which has several mathematical advantages). Two other axiomatizations that I mention do not use functions on a measure space at all.

On the history of development of Kolmogorov's axiomatization, there is a paper:

Jack Barone and Albert Novikoff, A History of the Axiomatic Formulation of Probability from Borel to Kolmogorov: Part I, Archive for History of Exact Sciences , 8.III.1978, Vol. 18, No. 2 pp. 123-190

But it covers only history of Kolmogorov's axiomatization, most alternative approaches are out of scope. A paper with wider coverage is Glenn Shafer and Vladimir Vovk, The Sources of Kolmogorov’s Grundbegriffe.

Remark. It is very common that in popular history of science one person obtains all credit for a great advance, and contributions of all other participants are forgotten. Let me cite Maurice Frechet:

"It was at the moment when Borel introduced this new kind of additivity in 1909 that all the elements needed to formulate the whole body of axioms of probability theory came together. It is not enough to have all ideas in mind... one must make sure that their totality is sufficient, bring them together explicitly and take the responsibility for saying that nothing further is needed in order to construct the theory. This is what Kolmogorov did. This is his achievement. And we do not believe that he wanted to claim any others, so far as the axiomatic theory is concerned."

• You say "random variable" is an old term, but the oldest quotes I have seen are from ~1910 onwards, while the theory of probability is much older. Do you know earlier uses of the term? Jan 3 at 8:18
• @Michael Bachtold: I am sure that it is much older, and instead of searching references can confirm this by Google Ngram. Jan 3 at 15:22