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.

中文翻译：

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从不同组的优惠券混合中取样

收集器从包含 $g$ \textit{uniform} 优惠券组的池中采样有替换的优惠券，其中“统一组”意味着该组中的所有优惠券均等可能出现。对于每个 $j = 1, \dots, g$ 设 $T_j$ 为检测组 $j$ 所需的试验次数，即收集属于它的所有 $M_j$ 优惠券至少一次。我们推导出第 $l$ 个组是第一个被检测到的概率的显式公式（象征性地，$P\{T_l = \bigwedge_{j=1}^g T_j\}$）。我们还在 $g=2$ 的情况下计算此概率的渐近性，因为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 $T := T_1 \vee T_2$，即收集池中所有优惠券所需的试验次数（至少一次）。我们确定 $E[T]$ 和 $V[T]$ 的渐近线，