Polya, Inverse Polya, and Circular Polya Distributions of Order k for l-Overlapping Success Runs
Polya, Inverse Polya, and Circular Polya Distributions of Order k for l-Overlapping Success Runs
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l 重叠成功运行的 k 阶 Polya、逆 Polya 和圆形 Polya 分布
DOI：
10.1080/03610920601033942
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Z. Psillakis
中科院分区：
文献类型：
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作者：
F. S. Makri;Andreas N. Philippou;Z. Psillakis
考虑Polya-Eggenberger抽样方案，即，从包含W个白色（成功）球和B个黑色（失败）球的骨灰盒中随机抽取一个球，观察其颜色，然后将其沿着与抽取的球相同颜色的S个附加球一起返回骨灰盒。该方案被重复n次，并且N n，k，l，s表示长度为k的l-重叠成功运行的数量。如果画出的球排列在一个圆上，则长度为k的l-重叠成功运行的数量表示为。最后，W r，k，l，s表示根据Polya-Eggenberger采样方案直到第r次出现长度为k的l-重叠成功游程为止的绘图数量。对于长度为k的l-重叠成功游程，引入了k阶Polya分布、逆Polya分布和循环Polya分布，分别作为Nn，k，l，s，Wr，k，l，s和.这些分布包括已知的特殊情况和k阶的新分布。导出了它们的概率分布函数和平均值的精确公式，推广了已知的k阶分布的一些结果。新的分布的渐近结果也给，有关他们，分别为二项，负二项，循环二项分布的顺序k为l-重叠成功运行的长度k。
The Polya-Eggenberger sampling scheme is considered, i.e., a ball is drawn at random from an urn containing w white (success) balls and b black (failure) balls, its color is observed, and then it is returned to the urn along with s additional balls of the same color of the ball drawn. This scheme is repeated n times, and N n, k, l, s denotes the number of l-overlapping success runs of length k. If the balls drawn are arranged on a circle, the number of l-overlapping success runs of length k is denoted by . Finally, W r, k, l, s denotes the number of drawings according to the Polya-Eggenberger sampling scheme until the rth occurrence of the l-overlapping success run of length k. Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs of length k are introduced as the distributions, respectively, of N n, k,l, s , W r, k, l, s , and . These distributions include as special cases known and new distributions of order k. Exact formulae are derived for their probability distribution functions and means, which generalize several results on well-known distributions of order k. Asymptotic results of the new distributions are also given, relating them, respectively, to the binomial, negative binomial, and circular binomial distributions of order k for l-overlapping success runs of length k.