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A poisson limit law for a generalized birthday problem

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  • Henze, Norbert

Abstract

Balls are placed sequentially at random into n cells. Write Tn,c(m) for the number of balls needed until for the mth time a ball is placed into a cell already containing c - 1 balls, where m [greater-or-equal, slanted] 1 and c [greater-or-equal, slanted] 2 are fixed integers. For fixed t> 0, let Xn,c denote the number of cells containing at least c balls after the placement of kn = [n1-1/c · t] balls. It is shown that, as n --> [infinity], the limit distribution of Xn,c is Poisson with parameter tc/c! As a consequence, the limit law of n1-c(Tn,c(m))c/c! is a Gamma distribution.

Suggested Citation

  • Henze, Norbert, 1998. "A poisson limit law for a generalized birthday problem," Statistics & Probability Letters, Elsevier, vol. 39(4), pages 333-336, August.
    • Handle: RePEc:eee:stapro:v:39:y:1998:i:4:p:333-336
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    References listed on IDEAS

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    1. Shigeru Mase, 1992. "Approximations to the birthday problem with unequal occurrence probabilities and their application to the surname problem in Japan," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(3), pages 479-499, September.
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    Cited by:

    1. Gerrit Gort & Wim J. M. Koopman & Alfred Stein, 2006. "Fragment Length Distributions and Collision Probabilities for AFLP Markers," Biometrics, The International Biometric Society, vol. 62(4), pages 1107-1115, December.

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