On the conditional and unconditional distributions of the number of success runs on a circle with applications

In this paper, we study the distributions of the number of success runs of length k and the number of success runs of length k given the number of successes in a sequence of independent and identically distributed (i.i.d.) binary trials arranged on a circle (circular sequence) based on three different enumeration schemes. The double generating functions, the probability functions and a formula for the evaluation of the higher order moments are given. Furthermore, we show that the results established in the case of an i.i.d. circular sequence can be extended to study the distribution of the number of success runs in a circular sequence of binary exchangeable trials. We offer tools for addressing the run-related problems arising from the circular exchangeable sequence. Some applications to practical problems such as reliability theory and Pólya urn models are given in order to show our theoretical results, which illustrate the potential use of run statistics. Finally, we address the parametric estimation in the distributions of the number of success runs.