Théorie analytique des probabilités

By P. Laplace. 1995

FIRST CHAPTER On the generating functions of one variable pp. 7–11 2. Let y x be any function whatever of x; if one forms the infinite series y 0 + y 1 t + y 2 t 2 + y 3 t 3 + · · · + y x t x + y x+1 t x+1 + · · · + y ∞ t ∞ , one can always imagine a function of t which, expanded according to the powers of t, gives this series: this function is that which I name generating function of y x. The generating function of any variable whatever y x is therefore generally a function of t which, expanded according to the powers of t, has this variable for coefficient of t x ; and reciprocally, the corresponding variable of a generating function is the coefficient of t x in the expansion of this function according to the powers of t; so that the exponent of the power of t indicates the rank that the variable y x occupies in the series , which one can imagine prolonged indefinitely on the left, relatively to the negative powers of t. It follows from these definitions that, u being the generating function of y x , that of y x+r is u t r ; because it is clear that the coefficient of t x in u t r is equal to that of t x+r in u; consequently it is equal to y x+r. The coefficient of t x in u 1 t − 1 is therefore equal to y x+1 − y x , or to the difference in the two consecutive quantities y x+1 and y x , a difference which we will designate by y x , being the characteristic of the finite differences. We have therefore the generating function of the finite difference of a variable quantity, by multiplying by 1 t − 1 the generating function of the quantity itself. The generating function of the finite difference of y x , a difference which we designate by 2 y x is thus u 1 t − 1 2 ; that of the finite difference of 2 y x , or 3 y x , is u 1 t − 1 3 , whence we can generally conclude that the generating function of the finite difference i y x is …

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Bayesian inversion for finite fault earthquake source models I—theory and algorithm (2013, background): “Pierre-Simon Laplace derived many important fundamental probabilistic inference results starting in 1774 and culminating in his treatise on probability (Laplace 1812).”
Ambiguity and Ambiguity Aversion (2014, background): “) Lehrer (2007) demonstrates that they do not violate the models of Lehrer (2009,2012), and Dillenberger and Segal (2012) demonstrate that they do not violate the model of Segal (1987a).”
Analytic Combinatorics (2009, background): “See Laplace’s treatise Théorie Analytique des Probabilit és[402] first published in 1812 for much fascinating mathematics related to this problem.”
Assessing Robustness of Intrinsic Tests of Independence in Two-Way Contingency Tables (2009, background): “A default prior for estimating the parameter p is usually chosen from one of the following distributions: the uniform (Bayes 1783; Laplace 1812), the Jeffreys prior (Jeffreys 1961; Bernardo 1979), Zellner’s prior (Zellner 1977), or that of Novik and Hall (1965).”