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EXCHANGEABLE PAIRS OF BERNOULLI RANDOM VARIABLES, KRAWTCHOUCK POLYNOMIALS, AND EHRENFEST URNS
Author(s) -
Diaconis Persi,
Griffiths Robert
Publication year - 2012
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2012.00654.x
Subject(s) - mathematics , binomial (polynomial) , bernoulli's principle , bivariate analysis , random variable , joint probability distribution , combinatorics , multinomial distribution , eigenfunction , distribution (mathematics) , discrete mathematics , pure mathematics , mathematical analysis , statistics , eigenvalues and eigenvectors , engineering , aerospace engineering , physics , quantum mechanics
Summary This paper derives characterizations of bivariate binomial distributions of the Lancaster form with Krawtchouk polynomial eigenfunctions. These have been characterized by Eagleson, and we give two further characterizations with a more probabilistic flavour: the first as sums of correlated Bernoulli variables; and the second as the joint distribution of the number of balls of one colour at consecutive time points in a generalized Ehrenfest urn. We give a self‐contained development of Krawtchouck polynomials and Eagleson’s theorem.