Abstract
Exact distributions of the numbers of failures, successes and successes with indices no less thanl (1≤l≤k−1) until the first consecutivek successes are obtained for some {0, 1}-valued random sequences such as a sequence of independent and identically distributed (iid) trials, a homogeneous Markov chain and a binary sequence of orderk. The number of failures until the first consecutivek successes follows the geometric distribution with an appropriate parameter for each of the above three cases. When the {0, 1}-sequence is an iid sequence or a Markov chain, the distribution of the number of successes with indices no less thanl is shown to be a shifted geometric distribution of orderk - l. When the {0, 1}-sequence is a binary sequence of orderk, the corresponding number follows a shifted version of an extended geometric distribution of orderk - l.
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This research was partially supported by the ISM Cooperative Research Program (92-ISM-CRP-16) of the Institute of Statistical Mathematics.
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Aki, S., Hirano, K. Distributions of numbers of failures and successes until the first consecutivek successes. Ann Inst Stat Math 46, 193–202 (1994). https://doi.org/10.1007/BF00773603
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DOI: https://doi.org/10.1007/BF00773603