论文标题
二进制搜索方案，用于确定所有受污染的标本
A binary search scheme for determining all contaminated specimens
论文作者
论文摘要
从$ n $不同的来源收集标本。每个标本的概率$ p $被污染（例如，在感染性疾病的情况下，$ p $是患病率），独立于其他标本。在许多情况下，小组测试是适用的，即可以从几个样本中取小部分，将它们混合在一起并测试混合物是否污染，因此，如果测试变成阳性，那么混合物中的至少一个样品将被污染。 在本文中，我们对二进制搜索方案进行了详细的概率分析，我们建议确定所有受污染的标本。更确切地说，如果应用此搜索方案，我们研究了为查找所有受污染标本所需的测试的数字$ t（n）$。我们得出了递归的，在某些情况下，我们为期望，差异和$ t（n）$的特征函数进行明确的公式。另外，我们将$ t（n）$的矩的渐近行为确定为$ n \ to \ infty $，从中我们获得了$ t（n）$（适当归一化）的限制分布，事实证明这是正常的。
Specimens are collected from $N$ different sources. Each specimen has probability $p$ of being contaminated (e.g., in the case of an infectious disease, $p$ is the prevalence rate), independently of the other specimens. In many cases group testing is applicable, namely one can take small portions from several specimens, mix them together and test the mixture for contamination, so that if the test turns positive, then at least one of the samples in the mixture is contaminated. In this paper we give a detailed probabilistic analysis of a binary search scheme, we propose, for determining all contaminated specimens. More precisely, we study the number $T(N)$ of tests required in order to find all the contaminated specimens, if this search scheme is applied. We derive recursive and, in some cases, explicit formulas for the expectation, the variance, and the characteristic function of $T(N)$. Also, we determine the asymptotic behavior of the moments of $T(N)$ as $N \to \infty$ and from that we obtain the limiting distribution of $T(N)$ (appropriately normalized), which turns out to be normal.
