Abstract
Let X 0, X 1, X 2,… be a sequence of {0, 1}-valued Markov chain. Let E 1 denote a run of “1” of length k and E 0 denote a run of “0” of length r. We observe which run comes sooner or later in the sequence X 0, X 1, X 2, …. The exact distributions of the numbers of overlapping sooner runs and nonoverlapping sooner runs until the later run occurs (for the first time) are derived. Let F 1 be a success run of length k or more and let F 0 be a failure run of length r or more. The exact distribution of the number of occurrences of the sooner event until the first occurrence of the later event between F 1 and F 0 is also studied. Further, when X’s have more than two values, more general problems are discussed and the exact joint distribution of the numbers of occurrences of the first, the second,…, and the j-th runs until the j-th run occurs (for the first time) is obtained in the case of independent trials.
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© 1997 Birkhäuser Boston
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Aki, S. (1997). On Sooner and Later Problems Between Success and Failure Runs. In: Balakrishnan, N. (eds) Advances in Combinatorial Methods and Applications to Probability and Statistics. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4140-9_22
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DOI: https://doi.org/10.1007/978-1-4612-4140-9_22
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8671-4
Online ISBN: 978-1-4612-4140-9
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