publication . Article . Part of book or chapter of book . 1956

A Combinatorial Lemma and its Application to Probability Theory

Frank Spitzer;
Open Access
  • Published: 01 Feb 1956 Journal: Transactions of the American Mathematical Society, volume 82, page 323 (issn: 0002-9947, Copyright policy)
  • Publisher: JSTOR
Abstract
To explain the idea behind the present paper the following fundamental principle is emphasized. Let X = (X 1,…, X n ) be an n-dimensional vector valued random variable, and let µ(x) =µ(x 1…, x n )be its probability measure (defined on euclidean n-space E n ). Suppose that X has the property that µ(x) =µ(gx) for every element g of a group G of order h of transformations of E n into itself. Let f(x) =f(x 1…, x n ) be a µ-integrable complex valued function on E n Then the expected value of f(x) is $$Ef\left( X \right) = \smallint f\left( x \right)d\mu \left( x \right) = \smallint \bar f\left( x \right)d\mu \left( x \right)$$ (1.1) , where $$\bar f\left( x \right) =...
Subjects
free text keywords: Applied Mathematics, General Mathematics, Cyclic permutation, Mathematical analysis, Expected value, Probability measure, Random variable, Probability theory, Rota–Baxter algebra, Mathematics, Topology, Lemma (mathematics), Combinatorics
Related Organizations
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue