In this paper, a simple and general method based on the finite Markov chain imbedding technique is proposed to determine the exact conditional distributions of runs and patterns in a sequence of Bernoulli trials given the total number of successes. The idea is that given the total number of successes, the Bernoulli trials are viewed as random permutations. Then, we extend the result to multistate trials. The conditional distributions studied here lead to runs and patterns-type distribution-free tests whose applications are widespread. Two applications are considered. First, a distribution-free test for randomness is applied to rainfall data at Oxford from 1858 to 1952. The second application is to develop runs and patterns-type distribution-free control charts which can be used as Phase I and/or Phase II control charts. Numerical results for two commonly used runs-type statistics, the longest run and scan statistics, are also given.

中文翻译：

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运行和模式的条件等待时间分布及其应用

在本文中，提出了一种基于有限马尔可夫链嵌入技术的简单而通用的方法，用于确定给定成功总数的伯努利试验序列中运行和模式的确切条件分布。这个想法是，给定成功的总数，伯努利试验被视为随机排列。然后，我们将结果扩展到多状态试验。这里研究的条件分布导致运行和模式类型的无分布测试，其应用广泛。考虑了两个应用程序。首先，对牛津从 1858 年到 1952 年的降雨数据进行无分布随机性检验。第二个应用是开发可用作阶段 I 和/或阶段 II 控制的运行和模式类型的无分布控制图图表。