When, by whom and in which paper where moments of a random variable used first in probability theory?


Long before Pearson, Chebyshev and his student A. A. Markov used moments to prove the Central Limit Theorem. The earliest paper of Chebyshev on this topic is dated 1887. But I do not claim that Chebyshev "was the first". On my opinion, a question of the type "who was the first" is meaningless and usually impossible to answer.


Chebyshev's original paper is called "On two theorems in probability", published in Russian In vol. 40 No 6 of the Imperial academy of sciences, French translation in Acta Math 14, 305-315. It is freely available in French in Chebyshev's Collected papers.

Remark. Chebyshev did not call them "moments" but used the name "integral residues".

  • $\begingroup$ Good point... disallowing the word “first” in question titles would encourage thoughtfulness. It’s nearly always more complicated. Grattan-Guinness: “One of the pitfalls in doing the history of mathematics is (...) accepting, as valid, questions of the form, “Who was the first to...?”.” $\endgroup$ – Francois Ziegler May 10 '18 at 2:50
  • $\begingroup$ Thank you for this answer :-) The point of asking who is the "first" is trying to find the oldest reference. How would you suggest such question be rephrased? $\endgroup$ – Julien__ May 10 '18 at 8:31
  • $\begingroup$ Oldest reference containing what? As I wrote, Chebyshev did not use the word "moment" but some other name. First and second moments (expectation and variance) were used long before (by the same Chebyshev in 1850s, for example), and so on. $\endgroup$ – Alexandre Eremenko May 10 '18 at 12:46
  • 1
    $\begingroup$ @AlexandreEremenko how is this different from your online post the first discovery about light, where you state : "It was Hero of Alexandria who first noticed this" ? Edit: interesting posts by the way $\endgroup$ – Julien__ May 11 '18 at 7:42
  • $\begingroup$ @Julien: I agree that my phrasing in that posting was imprecise. It would be more precise to say: "Hero of Alexandria already knew that..." $\endgroup$ – Alexandre Eremenko May 11 '18 at 13:19

My answer taken from the closed version of this question here

From MathWords

Moment was taken into Statistics from Mechanics by Karl Pearson when he treated the frequency-curve (or observation curve) as the sheet enclosed by the curve and the horizontal axis. See his "Asymmetrical Frequency Curves," Nature October 26th 1893: "Now the centre of gravity of the observation curve is found at once, also its area and its first four moments by easy calculation." [OED].


The phrase method of moments was used in a statistics sense in the first of Karl Pearson’s "Contributions to the Mathematical Theory of Evolution," (Philosophical Transactions of the Royal Society A, 185, (1894), p. 75.). Pearson used the method to estimate the parameters of a mixture of normal distributions. For several years Pearson used the method on different problems but the name only gained general currency with the publication of his 1902 Biometrika paper "On the systematic fitting of curves to observations and measurements" (David 1995). In "On the Mathematical Foundations of Theoretical Statistics" (Phil. Trans. R. Soc. 1922), Fisher criticized the method for being inefficient compared to his own maximum likelihood method (Hald pp. 650 and 719).


Moment generating function. R. A. Fisher seems to have brought this term into English in his "Moments and Product Moments of Sampling Distributions.," Proceedings of the London Mathematical Society, Series 2, 30, (1929), p. 238. He probably took the term from V. Romanovsky "Sur Certaines Éspérances Mathématiques et sur l'Erreur Moyenenne du Coefficient de Corrélation, Comptes Rendus, 180, (1925), 1897-1899. Romanovsky refers to "la function génératrice des moments" (p. 1898).


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