On Von Schelling Formula for the Generalized Coupon Collector Problem

TL;DR

This paper provides an algebraic proof of Von Schelling's formula for the coupon collector problem with non-uniform distributions, introducing a new theorem on sums of powers of subset probabilities and using binomial coefficients for decomposition.

Contribution

It offers a novel algebraic proof of Von Schelling's formula and introduces a new theorem on sums of powers of subset probabilities.

Findings

Proof of Von Schelling's formula for non-uniform distributions

Introduction of a new theorem on sums of powers of subset probabilities

Use of binomial coefficients for sum decomposition

Abstract

This paper gives an algebraic proof of the correctness of Von Schelling formula for the probability of the coupon collector problem waiting time for non-uniform distributions and partial collections. It introduces a theorem on sums of powers of subset probabilities which to our knowledge is new. A set of binomial coefficients is used as a basis for decomposition of these sums of powers.