AISM 52, 767-777
(Received April 21, 1998; revised February 22, 1999)
Abstract. A new distribution called a generalized binomial distribution of order $k$ is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer $\mu$ less than the specified length of the runs, an enumeration scheme called $\mu$-overlapping way of counting is defined. Let $k$ and $\ell$ be positive integers satisfying $\ell < k$. Based on independent Bernoulli trials, it is shown that the number of $(\ell -1)$-overlapping occurrences of success-run of length $k$ until the $n$-th overlapping occurrence of success-run of length $\ell$ follows the generalized binomial distribution of order $(k- \ell)$. In particular, the number of non-overlapping occurrences of success-run of length $k$ until the $n$-th success follows the generalized binomial distribution of order $(k-1)$. The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order $k$ is also given.
Key words and phrases: Binomial distribution of order $k$, Markov chain, probability generating function, stopping time, success-run, waiting time.
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