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On the Number of i.i.d. Samples Required to Observe All of the Balls in an Urn

Author

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  • Brad C. Johnson

    (University of Manitoba)

  • Thomas M. Sellke

    (Purdue University)

Abstract

Suppose an urn contains m distinct balls, numbered 1,...,m, and let τ denote the number of i.i.d. samples required to observe all of the balls in the urn. We generalize the partial fraction expansion type arguments used by Pólya (Z Angew Math Mech 10:96–97, 1930) for approximating $\mathbb{E}(\tau)$ in the case of fixed sample sizes to obtain an approximation of $\mathbb{E}(\tau)$ when the sample sizes are i.i.d. random variables. The approximation agrees with that of Sellke (Ann Appl Probab 5(1):294–309, 1995), who made use of Wald’s equation and a Markov chain coupling argument. We also derive a new approximation of $\mathbb{V}(\tau)$ , provide an (improved) bound on the error in these approximations, derive a recurrence for $\mathbb{E}(\tau)$ , give a new large deviation type result for tail probabilities, and look at some special cases.

Suggested Citation

  • Brad C. Johnson & Thomas M. Sellke, 2010. "On the Number of i.i.d. Samples Required to Observe All of the Balls in an Urn," Methodology and Computing in Applied Probability, Springer, vol. 12(1), pages 139-154, March.
  • Handle: RePEc:spr:metcap:v:12:y:2010:i:1:d:10.1007_s11009-008-9095-1
    DOI: 10.1007/s11009-008-9095-1
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    References listed on IDEAS

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    1. John E. Kobza & Sheldon H. Jacobson & Diane E. Vaughan, 2007. "A Survey of the Coupon Collector’s Problem with Random Sample Sizes," Methodology and Computing in Applied Probability, Springer, vol. 9(4), pages 573-584, December.
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    Cited by:

    1. James C. Fu & Wan-Chen Lee, 2017. "On coupon collector’s and Dixie cup problems under fixed and random sample size sampling schemes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(5), pages 1129-1139, October.
    2. Tung-Lung Wu, 2013. "On Finite Markov Chain Imbedding and Its Applications," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 453-465, June.

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