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

Question

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.

Answer 1

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."