Abstract
The classical occupancy problem is extended to the case where two types of balls are thrown. In particular, the probability that no urn contains both types of balls is studied. This is a birthday problem in two groups of boys and girls to consider the coincidence of a boy's and a girl's birthday. Let N 1 and N 2 denote the numbers of balls of each type thrown one by one when the first collision between the two types occurs in one of m urns. Then N 1 N 2/m is asymptotically exponentially distributed as m tends to infinity.
This problem is related to the security evaluation of authentication procedures in electronic message communication.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Davies, D. W. and Price, W. L. (1980). The application of digital signatures based on public key cryptosystems, Proc. 5th Internat. Symp. Comput. Commun., IEEE, Oct. 27–30, 1980, 525–530.
Fang, K.-T. (1985). Occupancy problems, Encyclopedia of Statistical Sciences, (eds. S. Kotz and N. L. Johnson), 6, 402–406, Wiley, New York.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York.
Hirano, K. (1986). Rayleigh distribution, Encyclopedia of Statistical Sciences, (eds. S. Kotz and N. L. Johnson), 7, 647–649, Wiley, New York.
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Applications, Wiley, New York.
Jordan, C. (Koroly) (1950, 1960). Calculus of Finite Difference, Chelsea Publishing Co., New York.
Knuth, D. E. (1967–1981). The Art of Computer Programming, Vol. 1–3, Addison-Wesley, Reading, Massachusetts.
Kolchin, V. F., Sevast'yanov, B. A. and Chistyakov, V. H. (1978). Random Allocation, (tr. ed. A. V. Balakrishna), V. H. Wistons and Sons, Washington, D.C.
Mueller-Schloer, C. (1983). DES-generated checksums for electronic signatures, Cryptologia, 7, 257–273.
Nishimura, K. and Sibuya, M. (1987). Probability to meet in the middle, KSTS-RR/87-006, Department of Mathematics, Keio University.
Popova, T. Y. (1968). Limit theorems in a model of distribution of particles of two types, Theory Probab. Appl., 13, 511–516.
Riordan, J. (1958). An Introduction to Combinatorial Analysis, Wiley, New York.
Sibuya, M. (1986). Stirling family of probability distributions, Japan J. Appl. Statist., 15, 131–146 (in Japanese).
Author information
Authors and Affiliations
About this article
Cite this article
Nishimura, K., Sibuya, M. Occupancy with two types of balls. Ann Inst Stat Math 40, 77–91 (1988). https://doi.org/10.1007/BF00053956
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00053956