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Who is the first person defined the concept of a random variable?

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    $\begingroup$ Possible duplicate of Who introduced random variables into probability? $\endgroup$
    – Conifold
    Jun 10, 2019 at 7:11
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    $\begingroup$ It is not a duplicate. The other entry is about the object; this one is about the term itself. What we currently call "random variables" were used much before the denomination became popular. $\endgroup$ Nov 29, 2022 at 19:28
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    $\begingroup$ One should distinguish between "who first used the expression random variable" and "who first gave the modern definition of the concept as a map of type Ω→ℝ ". The current answers don't seem to address the latter question. $\endgroup$ Jan 9 at 12:05

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The interesting material from mathwords

Random variable is found in 1914 in Biometrika: “nDx and nDy are now random variables independent of time.” [OED]

Variabile casuale is found in 1916 in F. P. Cantelli, “La Tendenza ad un limite nel senso del calcolo delle probabilità,” Rendiconti del Circolo Matematico di Palermo, 41, 191-201 (David, 1998). A. N. Kolmogorov used the term zufällige Gröβe in the Grundbegriffe der Wahrscheinlichkeitsrechnung (1933).

Random variable is found in 1934 in A. Winter, “On Analytic Convolutions of Bernoulli Distributions,” American Journal of Mathematics, 56, 659-663 and more visibly in H. Cramér’s Random Variables and Probability Distributions (1937) (David, 1998). Other, perhaps better, terms, including chance variable in Doob Annals of Mathematical Statistics, 6, (1935), p. 160 and stochastic variable in Wald & Wolfowitz, Annals of Mathematical Statistics, 10, (1939), p. 106 did not survive. J. L. Doob recalled the time when he was writing Stochastic Processes and W. Feller was writing his Introduction to Probability Theory and its Applications:

I had an argument with Feller. He asserted that everyone said “random variable” and I asserted that everyone said “chance variable.” We obviously had to use the same name in our books, so we decided the issue by a stochastic procedure. That is, we tossed for it and he won. From “A Conversation with Joe Doob,” Statistical Science 1997 (p. 307) Project Euclid.

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The reference above to Mathwords should be updated. In fact, Cantelli used the term "random variable" (variabilie casuale, in the original Italian) in his Sulla differenza media con ripetizione (1913), which could well be the first appearance in the history of probability theory. Cantelli's work cites (and abounds in results contained in) Variabilità e Mutabilittà, from 1912, a well-known work by the also well-known Gini. But the term in question is not used in it. It is a work that deals with the measurement of the variability of random variables, but at no time does it give them a specific name. Chebyshev's Des Valeurs Moyennes (1867), also refers to them only generically, "quantités".

The term gained some popularity in Italian and was used by Castelnuovo in his Calcolo delle Probabilità (1919). From there, it extended to French (Borel, Lévy), and finally to the English speaking world in the 1930's. It is relevant here to mention how Lévy praises Castelnuovo's work in his Calcul des Probabilités (1925). In it, the "event variable" is used in the way we are used to for "random variable" today. The terms variable éventuelle and variable aléatorire are found in other works of the time, such as Borel's treatises. Borel has a book on probability theory, Le Hasard (1914) where he does not use the term, but in each of two treatises from around the middle of the following decade he uses both of them.

On the other hand, Gosset in The Elimination of Spurious Correlation due to Position in Time or Space (1914) only juxtaposes the words "variable" and "random" in succession. But he never says anything like "such a random variable" or "study of random variables", as became common later. In Wintner's article On Analytic Convolutions of Bernoulli Distributions (1934), one of the first to use the combination as a term in English, the author generally uses a practically modern nomenclature. And not by chance, the article abounds in results published for the first time, as it says, in Lévy's Calcul. As Google Ngram shows, the term "random variable" only began to become popular in English in the 1930s.

I wrote a longer account of the story (in Spanish!) here.

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  • $\begingroup$ This is very interesting, but notice that even though Cantelli used the term "variabilie casuale" and gave a sort of definition, his definition is not the modern definition of a random variable being a map $X:\Omega \to \mathbb{R}$. Do you know who was the first to define random variables in this modern way? Kolmogorov? $\endgroup$ Jan 9 at 9:50

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