Probabilities of competing binomial random variables
Abstract
Suppose you and your friend both do n tosses of an unfair coin with probability of heads equal to α. What is the behavior of 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 phenomenons as parameters d, r and α vary. Our main tools are integral representations based on Fourier analysis.
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