URN Models of Correlation and a Comparison with the Multivariate Normal Integral

Abstract

In a special case of Polya's urn scheme, the probability that the first $n$ draws are all of the same color is interpreted as a function of the (single) correlation coefficient. A more general urn model is introduced in which the correlation between pairs of results may differ from pair to pair, and again the probability of consecutive colors is considered. This result is compared with the probability of coincidence in sign under the multivariate normal distribution. The comparison suggests a new approximation for the probability in the multivariate normal case. This approximation appears to be useful only in the Polya case, where the correlations are all equal.