Coupon collector's probabilities and generating function: analytic approach

TL;DR

This paper presents an analytic approach using recurrence differential equations to derive explicit probabilities and generating functions for the coupon collector's problem, connecting it to Polya Urn models.

Contribution

It introduces a novel analytic method based on recurrence differential equations to analyze coupon collector's probabilities, expanding beyond traditional combinatorial solutions.

Findings

Derived explicit probabilities for coupon collector's problem

Obtained generating functions using differential equations

Linked the approach to Polya Urn models

Abstract

Despite the coupon collector's problem has simple probabilistic solution using inclusion/exclusion principle \cite{PolyaUrn}, starting from a particular type of recurrence differential equation it is used an analytic approach to recover explicit probabilities and generating function. Note that this kind of recurrence differential equations often appear in Polya Urn context \cite{PolyaUrn}.