Joint distributions of successes, failures and patterns in enumeration problems
Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number S n of successes (and F n of failures) and the number X n of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time T r till the rth occurrence of the pattern and the number S T r of successes (and F T r of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (X n ,S n ), (T r ,S T r ) (and (X n ,S n ,F n ),(T r ,S T r ,F T r )) when X n belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.