Waiting for Scans Containing Two Successes

In the present chapter, we present a review of results pertaining to the distribution of waiting times for the occurrence(s) of scans of type 2∕r in sequences of binary trials. Our review covers the geometric distribution of order 2∕r, the negative binomial distributions of order 2∕r, and their generalizations. Exact and asymptotic results are presented and illustrated through numerical examples. In the case of the geometric distribution of order 2∕r, a new closed, exact formula is established. Several applications of the reviewed waiting time distributions in various scientific areas are discussed in some detail.