Probabilities of Competing Binomial Random Variables

Abstract

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