CURIOSITIES REGARDING WAITING TIMES IN POLYA'S URN MODEL
Norbert Henze, Mark Holmes
Transactions of A. Razmadze Mathematical Institute | Elsevier | Published : 2020
Consider an urn, initially containing b black and w white balls. Select a ball at random and observe its colour. If it is black, stop. Otherwise, return the white ball together with another white ball to the urn. Continue selecting at random, each time adding a white ball, until a black ball is selected. Let Tb,w denote the number of draws until this happens. Surprisingly, the expectation of Tb,w is infinite for the “fair” initial scenario b = w = 1, but finite if b = 2 and w = 109 . In fact, E[Tb,w] is finite if and only if b ≥ 2, and the variance of Tb,w is finite if and only if b ≥ 3, regardless of the number w of white balls. These observations extend to higher moments.
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Awarded by Australian Research Council
The work of MH is supported by Future Fellowship FT160100166 from the Australian Research Council. The authors thank a referee whose comments helped to improve the paper. The authors thank the organisers of the International Conference on Probability Theory and Statistics in Tbilisi, 2019, for bringing them together. Finally, the authors thank and congratulate Estate Khmaladze for many years of outstanding contributions to statistics and probability.