The Annals of Applied Probability

How Many IID Samples Does it Take to See all the Balls in a Box?

Thomas M. Sellke

Full-text: Open access

Abstract

Suppose a box contains $m$ balls, numbered from 1 to $m$. A random number of balls are drawn from the box, their numbers are noted and the balls are then returned to the box. This is done repeatedly, with the sample sizes being iid. Let $X$ be the number of samples needed to see all the balls. This paper uses Markov-chain coupling to derive a simple but typically very accurate approximation for $EX$ in terms of the sample size distribution. The approximation formula generalizes the formula found by Polya for the special case of fixed sample sizes.

Article information

Source
Ann. Appl. Probab., Volume 5, Number 1 (1995), 294-309.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004841

Digital Object Identifier
doi:10.1214/aoap/1177004841

Mathematical Reviews number (MathSciNet)
MR1325054

Zentralblatt MATH identifier
0823.60054

JSTOR
links.jstor.org

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Coupon collector's problem Markov chains Markov-chain coupling

Citation

Sellke, Thomas M. How Many IID Samples Does it Take to See all the Balls in a Box?. Ann. Appl. Probab. 5 (1995), no. 1, 294--309. doi:10.1214/aoap/1177004841. https://projecteuclid.org/euclid.aoap/1177004841


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