Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k
Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k
复制标题
直到某些停止时间规则和 k 阶广义二项分布之前指定长度的成功运行次数
DOI：
10.1023/a:1017585512412
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发表时间：
2000
影响因子：
1
通讯作者：
K. Hirano
中科院分区：
文献类型：
--
作者：
S. Aki;K. Hirano
定义了一种新的分布，称为广义二项分布，并研究了它的一些性质。严格研究了一类指定长度成功串的计数方案，包括非重叠和重叠计数方案。对于每个小于游程长度的非负整数μ，定义了一种称为μ-重叠计数法的枚举方案。令k和k是满足k <k的正整数。基于独立的伯努利试验，证明了长度为k的成功游程的（k − 1）-重叠出现次数直到第n次重叠出现长度为k的成功游程的次数服从（k− k）阶广义二项分布.特别地，长度为k的成功运行直到第n次成功的非重叠出现次数遵循阶数为（k− 1）的广义二项分布。即使底层序列从独立的伯努利试验序列改变为相关序列，如高阶马尔可夫相关试验，分布基本上保持不变。并给出了一个阶数广义二项分布的实例。
A new distribution called a generalized binomial distribution of orderkis 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 μ less than the specified length of the runs, an enumeration scheme called μ-overlapping way of counting is defined. Letkand ℓ be positive integers satisfying ℓ <k. Based on independent Bernoulli trials, it is shown that the number of (ℓ− 1)-overlapping occurrences of success-run of lengthkuntil then-th overlapping occurrence of success-run of length ℓ follows the generalized binomial distribution of order (k−ℓ). In particular, the number of non-overlapping occurrences of success-run of lengthkuntil then-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 orderkis also given.