从不同试样组的混合物中取样
Sampling from a mixture of different groups of coupons
摘要：
收集器从包含$g$ \textit{uniform}试样组的池中对具有替换的试样进行采样，其中“均匀组”意味着组中的所有试样都同样可能出现。对于每个$j = 1, \dots, g$，令$T_j$为检测组$j$所需的试验次数，即至少收集一次属于它的所有$M_j$试样。我们推导出一个明确的概率公式，$l$ -th组是第一个被检测到（象征性地，$P\{T_l = \bigwedge_{j=1}^g T_j\}$）。我们还计算了这种概率的渐近情况下$g=2$的优惠券的数量以某种方式增长到无穷大。然后，在两组的情况下，我们关注$T := T_1 \vee T_2$，即收集池中所有试样所需的试验次数（至少一次）。我们确定了$E[T]$和$V[T]$的渐近性，以及$T$的极限分布（适当归一化），因为试样的数量变得非常大。
英文摘要：
A collector samples coupons with replacement from a pool containing $g$ \textit{uniform} groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur. For each $j = 1, \dots, g$ let $T_j$ be the number of trials needed to detect Group $j$, namely to collect all $M_j$ coupons belonging to it at least once. We derive an explicit formula for the probability that the $l$-th group is the first one to be detected (symbolically, $P\{T_l = \bigwedge_{j=1}^g T_j\}$). We also compute the asymptotics of this probability in the case $g=2$ as the number of coupons grows to infinity in a certain manner. Then, in the case of two groups we focus on $T := T_1 \vee T_2$, i.e. the number of trials needed to collect all coupons of the pool (at least once). We determine the asymptotics of $E[T]$ and $V[T]$, as well as the limiting distribution of $T$ (appropriately normalized) as the number of coupons becomes very large.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
