发表机构
Department of Mathematics, University of Oslo（奥斯陆大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究从一副牌中抽到第一张A所需牌数的分布与期望，推广至多次出现及Dirichlet分布，并开发了在未知牌组或总体大小时进行估计与置信推断的方法。
AI 中文摘要
从一副牌中，我需要抽取多少张牌才能抽到第一张A？我确定了这个等待时间$T$的分布，及其令人满意的期望值${\ m E}\,T=(N+1)/(n+1)$，其中$N$是牌的总数，$n$是A的数量；因此标准设置下为$53/5=10.6$。在解决了这个第一个问题之后，我继续探讨某些替代解法和扩展，例如涉及Beta近似。我还考虑了第2次、第3次、第4次出现A的分布和均值，并进行了推广，其中有一个Dirichlet分布等待着我们，并与均匀分布的顺序统计量有进一步的联系。此外，我开发了一种工具，用于在已知A的数量$n$但不知道牌组大小$N$的应用中获得估计量和完整的置信分布；相应地，也用于在$n$已知时对未知总体大小$N$进行推断。如果你在一个房间里有1000人，需要面试其中11人直到找到第一个左撇子，那么房间里有多少左撇子——这里我们既需要估计值，也需要一个清晰的不确定性度量。
英文摘要
From a deck of cards, how many cards do I need to draw, until the first ace? I identify the distribution for this waiting time $T$, and its satisfyingly nice expected value ${\rm E}\,T=(N+1)/(n+1)$, with $N$ the number of cards and $n$ the number of aces; hence $53/5=10.6$ for the standard setup. After having solved this Question One I go on to certain alternative solutions and extensions, involving e.g. Beta approximations. I also consider the distributions and means for the 2nd, the 3rd, the 4th occurrences of aces, with generalisations, where there is a Dirichlet distribution in wait for us, with further links to order statistics for the uniform. Furthermore, an apparatus is developed for obtaining estimators and full confidence distributions for applications where one knows the number $n$ of aces, but not the deck size $N$; and correspondingly for inference about the unknown population size $N$ when $n$ is known. If you have 1000 people in a room, and need to interview 11 of them until you've found the first left-handed person, how may left-handed are there in the room -- here we need both an estimate and a clear measure of uncertainty.
Comments10 pages, 3 figures. Statistical Research Report, Department of Mathematics, University of Oslo; will be submitted for publication
