# Probabilities of competing binomial random variables

Wenbo V. Li, Vladislav V. Vysotsky

Research output: Contribution to journalArticle

3 Citations (Scopus)

### Abstract

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.

Original language English (US) 731-744 14 Journal of Applied Probability 49 3 https://doi.org/10.1239/jap/1346955330 Published - Sep 2012

### Fingerprint

Random variable
Unfair
Fourier Analysis
Integral Representation
Monotonicity
Convexity
Phase Transition
Vary
Demonstrate
Random variables
Integral
Phase transition

### Keywords

• Binomial random variable
• Coin tossing
• Competing random variables
• Number of successes
• Phase transition
• Probability of winning

### ASJC Scopus subject areas

• Mathematics(all)
• Statistics and Probability
• Statistics, Probability and Uncertainty

### Cite this

Probabilities of competing binomial random variables. / Li, Wenbo V.; Vysotsky, Vladislav V.

In: Journal of Applied Probability, Vol. 49, No. 3, 09.2012, p. 731-744.

Research output: Contribution to journalArticle

Li, Wenbo V. ; Vysotsky, Vladislav V. / Probabilities of competing binomial random variables. In: Journal of Applied Probability. 2012 ; Vol. 49, No. 3. pp. 731-744.
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