Geoff Eagleson's (1969) characterization of bivariate Binomial (N,p) distributions of exchangeable random variables (X,Y) which have Krawtchouk polynomial eigenfunctions is that the eigenvalues of the distribution have the form, for p greater than or equal to 1/2, as a mixture of Krawtchouk polynomials, scaled to be unity at zero. We show that this characterization is equivalent to...