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Probabilities of Competing Binomial Random Variables

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

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.

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