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