Sampling from a Mixture of Different Groups of Coupons

Abstract

A collector samples coupons with replacement from a pool containing g uniform groups of coupons, where “uniform group ” means that all coupons in the group are equally likely to occur (while coupons of different groups have different probabilities to occur). For each j = 1,…, 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 first derive formulas for the probabilities P{T₁ < … < T_g} and $P\left\{{{T_1} = \wedge _{j = 1}^g\,{T_j}} \right\}$. After that, without severe loss of generality, we restrict ourselves to the case g = 2 and compute the asymptotics of P{T₁ < T₂} as the number of coupons grows to infinity in a certain manner. Then, we focus on T:= T₁ ∨ T₂, i.e. the number of trials needed to collect all coupons of the pool (at least once), and 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 large.