### 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) |
---|---|

Pages (from-to) | 731-744 |

Number of pages | 14 |

Journal | Journal of Applied Probability |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2012 |

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### 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

*Journal of Applied Probability*,

*49*(3), 731-744. https://doi.org/10.1239/jap/1346955330

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

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 49, no. 3, pp. 731-744. https://doi.org/10.1239/jap/1346955330

}

TY - JOUR

T1 - Probabilities of competing binomial random variables

AU - Li, Wenbo V.

AU - Vysotsky, Vladislav V.

PY - 2012/9

Y1 - 2012/9

N2 - 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.

AB - 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.

KW - Binomial random variable

KW - Coin tossing

KW - Competing random variables

KW - Number of successes

KW - Phase transition

KW - Probability of winning

UR - http://www.scopus.com/inward/record.url?scp=84872202907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872202907&partnerID=8YFLogxK

U2 - 10.1239/jap/1346955330

DO - 10.1239/jap/1346955330

M3 - Article

AN - SCOPUS:84872202907

VL - 49

SP - 731

EP - 744

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 3

ER -