Polya, Inverse Polya, and Circular Polya Distributions of Order k for l-Overlapping Success Runs

Abstract

The Polya-Eggenberger sampling scheme is considered, i.e., a ball is drawn at random from an urn containing w white (success) balls and b black (failure) balls, its color is observed, and then it is returned to the urn along with s additional balls of the same color of the ball drawn. This scheme is repeated n times, and N _{n, k, l, s} denotes the number of l-overlapping success runs of length k. If the balls drawn are arranged on a circle, the number of l-overlapping success runs of length k is denoted by . Finally, W _{r, k, l, s} denotes the number of drawings according to the Polya-Eggenberger sampling scheme until the rth occurrence of the l-overlapping success run of length k. Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs of length k are introduced as the distributions, respectively, of N _{n, k,l, s} , W _{r, k, l, s} , and . These distributions include as special cases known and new distributions of order k. Exact formulae are derived for their probability distribution functions and means, which generalize several results on well-known distributions of order k. Asymptotic results of the new distributions are also given, relating them, respectively, to the binomial, negative binomial, and circular binomial distributions of order k for l-overlapping success runs of length k.

Keywords:

• Binomial
• Circular
• Distributions of order k
• Inverse Polya
• l-overlapping
• Negative binomial
• Non overlapping
• Overlapping
• Polya
• Success runs

Mathematics Subject Classification:

• Primary 60C05
• Secondary 62E15

Acknowledgments

We wish to thank the Associate Editor and the referee of our article for their helpful suggestions which led to several improvements.