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".
My answer taken from the closed version of this question here
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).