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Probabilities of Competing Binomial Random Variables

Part of: Harmonic analysis in one variable Probability theory on algebraic and topological structures Limit theorems

Published online by Cambridge University Press:  04 February 2016

Wenbo V. Li  and
Vladislav V. Vysotsky
Wenbo V. Li*
Affiliation:
University of Delaware
Vladislav V. Vysotsky*
Affiliation:
Arizona State University, Steklov Mathematical Institute and St. Petersburg State University
*
∗ Postal address: Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE 19716, USA. Email address: wli@math.udel.edu
∗∗ Postal address: School of Mathematical and Statistical Sciences, Arizona State University, PO Box 871804, Tempe, AZ 85287-1804, USA. Email address: vysotsky@asu.edu
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Abstract

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Suppose that both you and your friend toss an unfair coin n times, for which the probability of heads is equal to α. What is the probability that you obtain at least d more heads than your friend if you make r additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of n, and demonstrate surprising phase transition phenomenon as the parameters d, r, and α vary. Our main tools are integral representations based on Fourier analysis.

Keywords

Binomial random variablenumber of successescompeting random variablesprobability of winningcoin tossingphase transition

MSC classification

Primary: 60B99: None of the above, but in this section 60F99: None of the above, but in this section 42A61: Probabilistic methods
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 731 - 744
Copyright
© Applied Probability Trust 

Footnotes

Supported in part by NSF grants DMS-0805929, NSFC-6398100, and CAS-2008DP173182.

The author began this work at the University of Delaware, and was supported in part by the grant NSh. 4472-2010-1.

References

Addona, V., Wagon, S. and Wilf, H. (2011). How to lose as little as possible. Ars Math. Contemp. 4, 29–62.CrossRefGoogle Scholar
Dong, Z., Li, W. V. and Song, C. (2011). Integral representations for binomial sums of chances of winning. In preparation.Google Scholar
Ghahramani, S. (2005). Fundamentals of Probability, 3rd edn. Prentice Hall.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Karlin, S. (1968). A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Lengyel, T. (2011). On approximating point spread distributions. J. Statist. Comput. Simul. 81, 1333–1344.CrossRefGoogle Scholar
Ross, S. (2010). A First Course in Probability, 8th edn. Prentice Hall.Google Scholar
Zwillinger, D. (ed.) (2002). CRC Standard Mathematical Tables and Formulae, 31st edn. Chapman and Hall/CRC.CrossRefGoogle Scholar