The two-color urn model: recursions for the moments

Abstract

For the two-color reinforcement-depletion urn model, with balancing reinforcement and depletion held constant over cycles, a recursive formula is given from which all factorial moments (for white balls, for example) can be determined. When the reinforcement of each color is positive, the stationary distribution of white balls (infinite number of cycles) turns out to be determined by three parameters. namely (i) the total number of balls in the urn, (ii) the richness of the reinforcement, or ratio of white ball reinforcement to total reinforcement, and (iii) the size of the white ball reinforcement. In addition, the distribution mimics the binomial (with less variance and skewness (Vj3i) ) and from formulas for the exact first four moments rapidly approaches normality. On the basis of the few cases studied, an approximating Gram-Charlier distribution with a binomial nucleus is only moderately successful.

Keywords:

• factorial moment generating function
• moment approximants
• symbolic recursions