Abstract:

Assuming that there are \(N\) types of coupons, where the probability that the \(i\)th coupon appears is \(p_i \geq 0\) for \(i = 1, \ldots, N\), with \( \sum_{i=1}^{N} p_i = 1 \) we consider the variable \(T_k\), which represents the number of acquisitions needed to obtain \(k \leq N\) different coupons, and the variable \(Y_n\), which represents the number of different coupons obtained in \(n\) acquisitions. In the coupon collector problem, it's of interest to obtain the expected value of these random variables, as well as their \(r\)th moments. We provide new expressions for the \(r\)th moments of \(T_k\) and \(Y_n\), and give expressions for their moment-generating functions. Unlike known formulas, our formula for the \(r\)th moment of \(T_k\) is given in terms of recursive expressions, while that of \(Y_n\) is given in terms of finite sums, allowing for easier computational implementation. Furthermore, our formulas enable obtaining simplified expressions for the first few moments of these variables.

2010 AMS Mathematics Subject Classification: Primary 60C05; Secondary 60E99.

Keywords and phrases: coupon collector problem, combinatorial probability, moments, moment generating function.